Quantum Solid-State Physics 9780387191034, 0387191038, 9783540191032, 3540191038

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7

Springer Series in Solid-State Sciences Edited by Peter Fulde

Springer Series in Solid-State Sciences Editors: M. Cardona, P. Fulde, K. von Klitzing, H.-J. Queisser

Managing Editor: H. K. V. Lotsch

Volumes | - 49 are listed on the back inside cover

50 Multiple Diffraction of X-Rays in Crystals

68 Phonon Scattering in Condensed Matter V

51 Phonon Scattering in Condensed Matter

69 Nonlinearity in Condensed Matter

52 Superconductivity in Magnetic and Exotic

70 From Hamiltonians to Phase Diagrams

By Shih-Lin Chang

Editors: W. Eisenmenger, K. La8mann, and S. Déttinger

Materials. Editors: T. Matsubara and A. Kotani

53 Two-Dimensional Systems, Heterostructures, and Superiattices Editors: G. Bauer, F. Kuchar, and H. Heinrich

54 Magnetic Excitations and Fluctuations

Editors: S. Lovesey, U. Balucani, F. Borsa, and V. Tognetti

55 The Theory of Magnetism II

Thermodynamics and Statistical Mechanics

By D.C. Mattis

56 Spin Fluctuations in Itinerant Electron Magnetism. By T. Moriya

57 Polycrystalline Semiconductors

Physical Properties and Applications

Editor: G. Harbeke

58 The Recursion Method and Its Applications Editors: D.G. Pettifor and D.L. Weaire

59 Dynamical Processes and Ordering on Solid Surfaces. Editors: A. Yoshimori and M. Tsukada

60 Excitonic Processes in Solids

By M. Ueta, H. Kanzaki, K. Kobayashi, Y. Toyozawa,

and E. Hanamura

61 Localization, Interaction, and Transport

Phenomena. Editors: B. Kramer, G. Bergmann, and Y. Bruynseraede

Editors: A.C. Anderson and J. P. Wolfe

Editors: A.R. Bishop, D.K. Campbell, P. Kumar, and S.E. Trullinger

The Electronic and Statistical-Mechanical Theory of sp-Bonded Metals and Alloys. By J. Hafner

71 High Magnetic Fields in Semiconductor Physics. Editor: G. Landwehr

72 One-Dimensional Conductors

By S. Kagoshima, H. Nagasawa, and T. Sambongi

73 Quantum Solid-State Physics

By S. V. Vonsovsky and M. 1. Katsnelson

74 Quantam Monte Carlo Methods in Equilibrium and Nonequilibrium Systems. Editor: M. Suzuki

75 Electronic Stracture and Optical Properties of Semicoconductors By M.L. Cohen and J. R. Chelikowsky

76 Electronic Properties of Conjugated Polymers Editors: H. Kuzmany, M. Mehring, and S. Roth

T1 Fermi Surface Effects

Editors: J. Kondo and A. Yoshimori

78 Group Theory and Its Applications in Physics By T. Inui, Y. Tanabe, and Y.Onodera

79 Elementary Excitations in Quantum Fluids Editors: K.Ohbayashi and M. Watabe

80 Mente Carlo Simulation in Statistical Physics An Introduction. By K. Binder and D. W. Heermann

62 Theory of Heavy Fermions and Valence

81 Core-Level Spectroscopy in Condensed Systems. Editors: J. Kanamori and A. Kotani

63 Electronic Properties of Polymers and Related

82 Introduction to Photoemission Spectroscopy

Fluctuations. Editors: T. Kasuya and T.Saso

Compounds. Editors: H. Kuzmany, M. Mehring, and S.Roth

64 Symmetries in Physics. Group Theory Applied to Physical Problems. By W. Ludwig and C. Falter

65 Phonons: Theory and Experiments II

Experiments and Interpretation of Experimental Results. By P. Briiesch

66 Phonons: Theory and Experiments III Phenomena Related to Phonons By P. Briiesch

67 Two-Dimensional Systems: Physics and New

Devices

Editors: G. Bauer, F. Kuchar, and H. Heinrich

By S.Hifner

83 Physics and Technology of Submicron Stractares. Editors: H. Heinrich, G. Bauer, and F. Kuchar

84 Beyond the Crystalline State

An Emerging Perspective By G. Venkataraman, D. Sahoo, and V. Balakrishnan

85 The Fractional Quantum Hall Effect

Properties of an Incompressible Quantum Fluid By T. Chakraborty and P. Pietilainen

86 The Quantum Statistics of Dynamic Processes By E. Fick, G. Sauermann

S. V. Vonsovsky

M.I. Katsnelson

Quantum Solid-State Physics With 151 Figures

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong

Academican Serghey V. Vonsovsky Dr. Mikhail I. Katsnelson Institute of Metal Physics, Ural Science Research Center of the USSR Academy of Sciences, 18, S. Kovalevskaya Street. SU-620219 Sverdlovsk, GSP-170/ USSR Series Editors:

Professor Dr., Dres. h.c. Manuel Cardona Professor Dr., Dr. h.c. Peter Fulde

Phys C C

Professor Dr., Dr. h.c. Klaus von Klitzing

as

Professor Dr. Hans-Joachim Queisser

| / v

Da

Managing Editor: Dr. Helmut K. V. Lotsch

—o

£)

ce

=

1 WNW

Max-Planck-Institut fiir Festkérperforschung. HeisenbergstraBe

D-7000 Stuttgart 80, Fed. Rep. of Germany

SS

Springer-Verlag, TiergartenstraBe 17, D-6900 Heidelberg. Fed. Rep. of Germany

Title of the original Russian edition: Kvantovaja fizika tverdogo tela, “Nauka™ Publishing House, Moscow 1983

ISBN 3-540-19103-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19103-8 Springer-Verlag New York Berlin Heidelberg Vonsovskil, S. V. (Sergel Vasil ‘evich) [Kvantovaia

fizika tverdogo tela. English] Quantum

solid-state physics/S.V. Vonsovsky, M.I. Katsnelson. p.cm. — (Springer scrics in solid-state sciences : 73). Translation of: Kvantovaia fizika tverdogo tela. Bibliography: p. Includes indexes.

ISBN 0-387-19103-8 (U.S.)

.

1. Solid state physics. 2. Quantum theory. I. Katsnel ‘son, M.I. (Mikhail III. Series. QC176.V6613 1989, 530.4’I-de 19 89-5948

losifovich) I. Title.

This work is subject to copyright. All rights are reserved, whether the whole or part of the

material is concerned, specifically the rights of translation. reprinting. reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thercof is only permitted under the provisions of the German Copyright Law of September 9, 1965. in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Printed in Germany

The use of registered names, trademarks, ctc. in this publication docs not imply. even in the absence of a specific statement. that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printing: Colordruck

Dorfi GmbH.

Berlin: binding: Helm, Berlin

2154/3155-543210- Printed on acid-free paper

Preface

The quantum theory of solids occupies a peculiar and important place in the general structure of modern theoretical physics. There are currently no grounds for questioning the statement that all properties of solids can, in principle, be accounted for on the

basis of firmly established principles of quantum and statistical mechanics. Neverthe-

less. these properties of real solids and the condensed state of matter in general, are

so complicated

and

diverse that it is well nigh

in other

fundamental

impossible,

at least at present,

to

explain rigorously and fully from first principles the observed characteristics of crystals, even those which are close to perfect, - let alone explain the fact that they exist! Therefore, alongside the mathematical methods and physical concepts applied more

areas

of

theoretical

physics,

solid-state

theory

has

developed approaches of its own to account for the most important properties of the various substances. Significantly, these approaches now have a profound reciprocal effect on not only statistical physics but also on particle physics, and even on astrophysics

and cosmology.

Apart from

this, the tremendous

and ever-increasing

applied significance of solid-state theory must be noted. Suffice it to mention here the theory of semiconducting devices, the theory of strength and plasticity, the theory of magnetic properties of materials, etc. In this respect, modern solid-state theory employs with great practical success a sufficiently simple and, at the same time, adequate theoretical background, based on a purely phenomenological approach and

microscopic models that are comparatively simple in terms of mathematics and very lucid physically. As stated above, a quantum theory of solids that realizes the “first-principles” program in its entirety, i.e., a theory in which all properties of solids are derived from those of individual constituent atoms, does not exist. However, it may well be assumed that, for example, the indubitable and sizable success of the pseudopotential method

that

now

enjoys wide

use in the theory

of simple

(normal)

metals

is an

important step toward the construction of such a physically consistent first principles theory. Rather than choosing the deductive method of presentation, we have therefore opted, in this text, for a method based on a treatment and analysis of simple empirically established properties of solids, resorting to more complicated models only where necessary. In a way, such an exposition reproduces the evolution of this important province of modern theoretical physics (differing in this respect from the diverse monographs and textbooks devoted to the problem concerned) and, in our view, is most appropriate to initiate the reader into the subject. We have also assumed that a detailed treatment of a number of classical topics such as the one-dimensional

VI

Preface

Schrédinger equation with a periodic potential, the metal-insulator criterion, the effect of electric and magnetic fields on electronic states, and other similar problems would be very instructive. At the same time, the book presents a number of up-to-

date topics: the scattering of neutrons by the crystal lattice, plasma and Fermi liquid effects, elements of pseudopotential theory, fundamentals of the theory of disordered

systems, etc. It is assumed that the essentials of quantum and statistical mechanics are a sufficient theoretical background for reading this text. As far as possible we have tried to outline the body of mathematics in conjunction with those specific problems in which it is immediately exploited. Thus, for instance, the secondary quantization method is expounded for the first time in connection with the problem of neutron scattering on phonons, the resolvent (Green’s-function) method is outlined in connec-

tion with the problem of electron localization on impurities, etc. The presentation of a number of problems, in which allowance for correlation effects in electronic systems

(superconductivity, the properties of transition metals, and the like) is essential, has turned out to be extremely concise, for we did not succeed in finding a simplified enough version of the mathematical sophistication needed for a more rigorous exposition.

In those (and some other) cases we had to confine ourselves to simple

model problems and often to purely qualitative lines of argument. Although

the present book outlines sufficiently general properties of solids, as

well as some specific problems of the physics of semiconductors, ionic crystals, etc., it is still primarily the metal that we view as the model of “a solid in general.” This is partly because the metal is in essence a gigantic molecule and demonstrates most dramatically the features peculiar to the electronic properties of crystals, which are not reducible to a mere “sum” of the properties of individual constituent atoms or molecules.

An acquaintance with this text should prepare the reader for a more detailed study

of particular areas of solid-state theory, which at present are outlined fully enough at

an up-to-date level in the abundant body of special literature. The text is intended to appeal to experimental physicists, chemists, engineers concerned with problems of solid-state theory and wishing to become conversant with the pertinent body of mathematics, theoretical physicists in other fields, and undergraduate and graduate students. Throughout the book the numerical values of physical quantities are given in both the units normally employed by physicists and SI units. electrodynamics the CGSE system is used throughout.

For

the

equations

of

This monograph evolved largely from notes written for a course offered in the

Department of Theoretical Physics of the Ural Institute for Physics and Engineering by Prof. S. P. Shubin as far back as the thirties and from lectures delivered for many years at the A.M. Gorky Ural State University by one of us (S. V. Vonsovsky).

Very encouraging assistance during the development of the manuscript was rendered by Dr. L. A. Shubina who executed a lot of systematizing work, helped to improve the text, and skillfully typed the manuscript in its entirety. We are deeply indebted to her for this invaluable help and express keen sorrow at her not being able to see the book published.

Preface

During

the

development

of the

English

edition,

numerous

alterations

Vu

were

introduced in the text. New material has been added: Sects 1.7 (and a large portion of Sect. 1.8), 1.10, 2.5, 4.2.5, 4.4.3, 4.6.7, 4.9, 5.3.2 (and a large portion of Sect. 5.3.3),

5.7. The inaccuracies noticed in the Russian edition have been corrected, a number of new figures added, and the list of references extended substantially.

It is our pleasant duty to thank Dr. H.Lotsch for his offer to have the book published and for making it a reality. We are sincerely grateful to Mr. A.P.

Zavarnitsyn

for the

English

translation

and

the

assistance

rendered

during

the

development of this project. We would like to express our sincere appreciation to Mrs. M.L. Katsnelson for her enthusiastic help in preparing the manuscript and to Mr. V.G. Reprintsev for typing the rough draft. Sverdlovsk, Spring 1989

Serghey V. Vonsovsky Mikhail I. Katsnelson

Contents

1. Introduction. General Properties of the Solid State of Matter

...........

1

1.1.

General Thermodynamic Description of the Solid State

1.2.

Crystal Structure of Solids...

1.6 1.7.

Qualitative Concepts of the Electronic and Nuclear Crystal Structure .. Fundamental Concepts of the Chemical Bonding in Solids .........

43 48

1.7.1

49

1.3 1.4 1.5

Structure of Solids...

0...

eee

eee

6.

eee

Interaction Between Atoms (Ions) with Filled Electron Shells

...

!

6

26 29 32

Molecular Orbitals. 2.0... ee eee The Heitler-London Method .................0..0004 Covalent Bond ......... 0.0.0.0... cee eee eee Electrostatic Bonding Energy of Ionic Crystals. ............

55 60 65 69

Types of Crystalline Solids ..........0 0.000.000 0.00000 0008 1.8.1 Tonic Crystals 2... 0... eee eee 1.8.2 Covalent Crystals and Semiconductors ................0.

22 73 74

1.8.3

Metals, Their Alloys, and Compounds

eee

78

1.8.5 1.8.6 1.8.7

Hydrogen-Bonded Crystals ..............00-000 000 ee Quasi-One-Dimensional and Quasi-Two-Dimensional Crystals .. Quantum Crystals... 2... eee

78 79 80

1.8.4 Molecular Crystals...

1.9

2... 0.0

Reciprocal Lattice... 2... eee Examples of Simple Crystal Structures... 2... .....002.2020..00. Experimental Techniques for Determining the Periodic Atomic

1.7.2 1.7.3 1.7.4 1.7.5 1.8

2...

...........

2.0...

ce

...............-..

ee

Formulation of the General Quantum-Mechanical the Crystal... 0...

1.10 Properties of Disordered Condensed Systems 1.10.1

General Remarks.

.........0.

76

Problem of eee

.................

0.20.0...

1.10.2 Metallic Glasses (Example of Amorphous Solids)

000000

..........

. Dynamic Properties of the Crystal Lattice 2...................00. 2.1 The Dynamics of the Ionic Lattice... 62... ee eee 2.1.1 A Linear Monatomic Array ...............0.0..00000.

2.1.2

A Linear Diatomic Array

2.1.3 2.1.4

The Three-Dimensional Crystal Case... 0.2. ..........2-Quantization of Ionic-Lattice Vibrations ...............-.

..........0.....

000000200005

85

90 90

92

97 97 97

101

105 109

Contents

2.2 2.3.

The Specific-Heat Capacity of the Lattice

.............

111

00000000005

120

2.3.2 Heat-Capacity Term Linear in Temperature .............. 2.3.3 Thermal Conductivity of an Ionic Lattice ................

122 123

2.4 2.5

Localization of Phonons on Point Defects .................04. Heat Capacity of Glasses at Low Temperatures ................

124 127

2.6

High-Frequency Permittivity of Ionic Crystals

131

2.7

Allowance for Anharmonic Terms.

................000.

2.3.1 Thermal Expansion of Crystals .............

Lattice Scattering and the Méssbauer Effect

2.7.1

0.000000 05

.................

..................

Scattering Probability and the Correlation Function

.........

2.7.2 Some Properties of the Phonon Operators and of the Averages Containing Them... 02... ee 2.7.3 Calculating the Dynamic Form Factor in the Harmonic Approximation

2...

ee

eee

120

134 134

138 143

2.7.4 Elastic Scattering... 6... 2 eee 2.7.5 Inelastic Scattering... 2... 6 ee 2.7.6 The Mossbauer Effect... 2.0... cee ee ee Conclusion... 2... 6.2 ee

147 149 152 155

3. Simple Metals: The Free Electron-Gas Model ..................... 3.1 Typesof Metals... 0.0.0... eee eee 3.2 Physical Properties of the Metallic State. Conduction Electrons ......

157

2.8

3.3. 3.4 3.5

Classical Conduction-Electron Theory (Drude-Lorentz Theory) ..... Itinerant Electron Theory According to Frenkel .............-... Application of Fermi-Dirac Quantum Statistics to

the Conduction-Electron Gas... 6... 6. eee 3.5.1 The Caseof T=OK............ 000000002

157 158 162 168

eee

170 173

3.5.2 The Low-Temperature Case (T >0 K, but T |r|, !), so that the A,Q and A,Q lines may

with high accuracy be considered parallel to one another and to the normal ¢. The path difference of two scattered rays is equal to A,B, —A,B,

=r-(t—t))=r-q.

(1.5.3)

544 (an by

A2 t 82

a

Fig. 1.24 Determination of the scattered-ray path

difference by two lattice sites, A, and A,

The vector (t — fo) = g, as seen from Fig. 1.25, is normal to a plane which, by convention, plays the role of a reflection plane for an incident (t,) and a reflected

(t) ray. If we denote the angle for the incident and reflected rays on this plane by 9, the scattering angle (the angle between ft) and ft) is equal to 29 and,

consequently, we find from Fig. 1.25

lq =It—tol =2sin9 .

(1.5.4)

36

1. Introduction. General Properties of the Solid State of Matter

Fig. 1.25. Construction of a normal to the X-ray glide plane in the crystal

If we introduce a wave vector k = 2zt/A, of which the modulus is equal to

|kl =2n/A,

(1.5.5)

the path difference (1.5.3) yields for the phase difference Ag =kr-(t—to) = k(r-g)

.

(1.5.6)

The amplitude of a scattered wave at point Q will be maximal for directions in which the phase difference 4g is a multiple of the quantity 2x, when the

amplitudes of the waves scattered from the atoms at sites A, and A, add constructively. Recall now that r is one of the translation vectors (1.2.9). Then the Laue conditions should hold for the diffraction maximum

00, = Ea; g) = 2am, (i= 1,2,3),

(1.5.7)

where m; are integers. Label the direction cosines of the vector g with respect to the a; axes by a;. Then (1.5.7) becomes

(a;-q) = 2a; sin9-a, = m;A .

(1.5.8)

This results in a selective diffraction picture of X-ray reflection from the crystal, because (1.5.8) is capable of solution only for some angles $ and wavelengths A with a given lattice (a;) and for the choice of the reflection plane (a; ). From (1.5.8) we now obtain the Wulf—Bragg relations for selective reflection from a given family of parallel crystal planes with a distance d between the

neighboring planes. It follows from (1.5.8) that the a; in the direction of the diffraction maximum are proportional to the quantities m,/a,, m,/a,, m3/a3.

The successive lattice planes (m, m2 m3) intercept the a, axes, according to the definition of the Miller indices, at the respective distances a, /m,, a,/m2,a3/m3. It follows from elementary geometry that the direction cosines to the planes (m, m,m;3) are also proportional to a;/m,; i.e., these planes are parallel to the reflection plane. The distance d (m, m,m,) between the adjacent planes of the family (m, m,m,) is equal to d(m, m, m3) = a, a,/m, = a,a,/m, = a3a3/m,

,

(1.5.9)

1.5

Experimental Techniques

37

and then the Laue equations take the form 2d(m,m,m;) sind =1 .

(1.5.10)

Observe that the integers m,, m,, and m, in (1.5.10) are not merely Miller indices of the different

crystal

planes,

for they

may

contain

an

integral

common

multiplier n (which cancels out due to Miller indexing). Therefore, if m; = m;,/n in

(1.5.10) are held to be Miller indices, we obtain

2d(m), mm)

sind = nd

.

(1.5.11)

This is the famous Wulf—Bragg formula in which the integer n gives the order of reflection. [The relation (1.5.11) was first derived by the Russian scientist G.W. Wulf in 1912, and a year later by the Englishman W. Bragg who verified it experimentally in 1913.] The Laue equations and the Wulf-Bragg formula are due only to the

fundamental

structure,

property

and

of the crystal, that is, the periodicity of its atomic

are associated

with

neither

its chemical

composition

nor

the

atomic arrangement in the reflection planes. The latter factors influence the

magnitude of peak intensity and are therefore very important in determining telative peak intensities. It is also seen from (1.5.11) that the condition for the

occurrence of a diffraction peak is 4 < 2d.

To describe X-ray diffraction in crystals, we make use of the concept of the reciprocal lattice introduced in Sect. 1.3. Instead of d(m', m'‘,m’,) from (1.3.8) we May introduce into (1.5.11) the quantity b(m’', mm’) '. Then, if g, and m; contain acommon multiplier n, the nth site of the reciprocal lattice (in the array of sites, counting from the origin of the reciprocal lattice) with the components g; (or m,) refers to an nth-order peak of an X-ray reflected from the corresponding crystal

planes. Each reciprocal-lattice site therefore refers to a possible reflection peak.

Ewald (1913) (see [1.19]) has offered a simple geometric interpretation for this

fact (Fig. 1.26).

e

.

.

.

.

Fig. 1.26. Geometric interpretation (after Ewald) of the relation between reflection peak and reciprocal-lattice site: O is the origin of coordinates of the reciprocal lattice; AO the incident wave vector; AB the reflected wave vector; OB the normal to reflection plane

38

1, Introduction. General Properties of the Solid State of Matter

Let the AO segment be equal to the vector t,/A of length A~! oriented along the normal to the incident radiation wave front. In precisely the same way, the AB segment is equal to the vector t/A of the same length, parallel to the direction along which the scattering is considered. Then the vector OB will be parallel to the scattering vector g (1.5.3) and normal to one of the lattice planes, e.g., g,. The length of OB, according to (1.5.4), is equal to 2sin(8/A), which, according to

(1.5.11), is equal to 1/d(g, 9293), if the diffraction condition is fulfilled. Thus OB lies in the direction of the reciprocal-lattice vector and is equal to it in modulus. Therefore, if the point O coincides with the origin of the reciprocal lattice, the

point B should occur at site g,. Then, following Ewald, we are in a position to execute the following construction. If O is the origin, we should draw a vector AO in the incident direction of length 4~', terminating at the origin, and construct a sphere of radius 4~' = |AO| with its center at A. In the case under consideration, this sphere should pass through point B. Therefore, the Laue (1.5.7) or Wulf-Bragg

(1.5.11) conditions are equivalent to the condition that a diffraction peak cannot

arise unless at least one more of the reciprocal lattice sites B joins this sphere. The latter is referred to as the propagation sphere. This method, replacing the

consideration of planes in direct space by points in reciprocal space, considerably simplifies the solution of all crystal diffraction problems. Since this method will later be used on more than one occasion, we present one more vector modification of the Wulf-Bragg conditions (1.5.11). Let

b* = 2n6, ,

(1.5.12)

where 6, is the reciprocal-lattice vector (henceforth the reciprocal-lattice vector

will be referred to as b}) and k is the wave vector of an incident wave (1.5.5). From the Ewald construction (Fig. 1.26) we find (40+ OB)? = (BA)’, or

(k + b*)? = (k)?. Hence (1.5.11) becomes 2(k-b*) + b*2

=0.

(1.5.13)

We now calculate the so-called atomic scattering factor [Ref. 1.12, Sect. 126]. Strictly speaking, to do this we need to use the methods of quantum mechanics. However, here we confine ourselves to a quasi-classical method. For simplicity,

we assume that the scattering process is elastic when the state of the scattering

atom does not change and so the potential energy of the atom may be averaged over its wave function. According to the Born approximation, the differential scattering cross section is proportional to the quantity \fur Vir) yar?

where ¥, and an elastically

= lVixgl?

’

y,, are respectively the wave functions of an incident scattered particle (|k| =|kol), ie, WY, =exp((i/h)p-r)

and and

1.5

Experimental Techniques

39

Wao = exp((i/h)po-r). Here p = hk and po = hk, are the momenta of the incident

and scattered particles. From the expression for V,,, we can readily obtain the

Wulf—Bragg formula (1.5.11). To this end, we expand the periodic potential V(r) in a Fourier series (1.3.2)

Vir) => Vyexp(ibz-r) , 6:

see also (1.5.12). For },, with wave functions as plane waves we then have

Ven = Dv SexpLi(ky + bF —k)-r]dr 53

__ [ Vyp at ko + BS —k = 0

(1.5.14)

~ )0 in all the other cases.

Thus, a scattered ray may be observed only in directions with the wave vectors k = ky + 5%. Apart from this, because of the elastic nature of the scattering, we have |&| =|ko|. From Fig. 1.26 it then follows immediately (setting t = k, to = ko) that |b3| = 2|k|sin 9. Next, using (1.5.4), (1.5.5), and (1.5.11) we arrive at (1.5.12). Returning to the scattering cross section, we see that here it is equal to the square of the modulus of the integral (1.5.5, 7):

fVirye"dr .

(1.5.15)

The average potential energy of an atom is equal to V(r) = eg(r), with g(r) being the atomic field potential and e the electronic charge. If we denote the

density of the electric charge of an atom by g(r), the g(r) and g(r) are related to each other by the Poisson equation Ag(r) = —4ze(r). The same equation should be satisfied by all the Fourier components of the potential ¢, exp (igr),

that is,

AL, exp(ig:r)] = q’ 9, exp (ig: r) = 4nQ, exp(ig-r) ; this immediately gives 9, = (41/47) 0,.

Substituting the explicit expressions for

the Fourier coefficients into this equation yields

fe(r)exp(—ig-r)dr = 4nq~? o(r)exp(—ig-r)dr .

(1.5.16)

The atomic charge density g(r) involves a nuclear point charge + Zed(r) [with Z

being the serial element number, 5(r) the Dirac delta function] and an electronic

40

1. Introduction. General Properties of the Solid State of Matter

charge Z of spatial density g(r). For the integral on the right-hand side of (1.5.16) we then obtain

fo(r)exp(—ig-r) = —efn(r)explig:r)dr + Ze . Thus the integral (1.5.15) has the form

JV (r)exp(ig-r)dr = 4ne*q”?[Z — F(q)] ,

(1.5.17)

where the quantity

F (q) = Jn(r)exp(ig-r)dr

(1.5.18)

is called the atomic form factor. It gives the magnitude of the amplitude of a wave scattered by the “smeared” electronic density n(r), to the amplitude of a wave scattered on a point electron; F(g) of the scattering angle 9 and of the wave vector of the particle being n(r) is a spherically symmetric function, i.e., n(r) = n(r), we have

ratio of the of the atom, is a function scattered. If

§n(r)exp(iq:r)dr = tht n(r)exp(igrcos$) 000

x r?sin9dSdedr = 4z { r?n(r)singr(qr)~ ‘dr

0

Regarding 4zr?n(r)dr = w(r)dr as the probability that the scattering electron resides in the layer between the spheres of radii r and r + dr, we get in place of

(1.5.18)

2

F(q) = § w(r)singr(qr)"‘dr .

0

(1.5.19)

Tables are available for F(q) which are calculated according to the Thomas-— Fermi method or the Hartree-Fock method [Ref. 1.12, Sect. 13] using the values of n(r) or w(r). Figure 1.27 presents plots of the neutron (N) and X-ray (X) form factors for iron as functions of q = 4zsin9/A [1.20]. With neutrons the quantity n(r) should signify the density of noncompensated magnetic moments, i.e., the charge-density difference for electrons with positive (+) and negative (—) spin projections: An(r) = n,(r) — n_(r). Only the spin-dependent part of the neutron form factor is implied here. Since the spin density is related chiefly to the outer portion of the electronic shell of the atom (3d shell in the atom of iron), the spin-density radial distribution maximum 4n(r) lies farther from the center

of the nucleus than does the charge-density maximum. Equation (1.5.19) shows that the magnitude of the magnetic form factor falls off faster with decreasing q

1.5

024

6 60., q,A

Experimental Techniques

41

Fig. 1.27. Neutron (N) and x-ray (X) formfactors F(q) of iron as a function of the wave number q determined by the change of momentum in an atom-scattered particle

than that of the X-ray form factor (Fig. 1.27). An immediate observation from

(1.5.19) is that the value of the form factor for q = 0 is equal to the number of electrons in the atom, Z (for the X-ray case), or to the magnitude of the noncompensated spin moment (in the neutron case). As has already been noted, the Laue formula as well as the Wulf—Bragg formula (with the assumptions that have been made) only allow us to determine the position of the peaks. But is it also important to know the magnitude of the

relative intensity of the various peaks, their shape, and their temperature dependence. That information would enable us to determine the type of crystal

unit cell, the number of and the arrangement of atoms in it, etc. To this end, it is necessary to determine the amplitude of a wave scattered in some direction by

all unit-cell atoms. Just as with the atomic scattering factor, we call the ratio of the amplitude of a reflected wave to the amplitude of a wave scattered on a point electron (provided that the particle being scattered is of the same wavelength) the structure amplitude F(g,g.9g3) for a g,-type reflection. The expression for F(g1 9293) has the form

F(g19293) = >, F.exp(id@,) = > F,exp(i(2x/A)R,9)

(1.5.20)

where the summation is taken over all unit-cell atoms, 4@, is the phase of a wave

scattered by an s-type atom (s = 1, 2,..., a), R, is the vector defined from (1.2.10), and F, is the atomic factor of an atom s (1.5.18, 19). In keeping with (1.5.6, 10), we have

Req = 49129 + 9229 +9569) ,

(1.5.21)

and therefore it follows from (1.5.20) that

F (919293) = >. Fsexp[i2n(g, £9 + 9269 + 93€9)]

(1.5.22)

42

1. Introduction. General Properties of the Solid State of Matter

and also that 2 IF(gi9293)I’ = [ E.cos ano, g

.

+ [ Erasinanta

tard

+000)

(1.5.23)

2 of +9269 +9369

|

:

If all of the unit-cell atoms are alike, the factors F, are the same for all s, and they

may be taken outside the summation sign. Instead of (1.5.22) we then obtain

(1.5.24)

F(9,9293) = FS , where the quantity S is called the structure factor and has the form

S = Yexp[i2a(g, 2? + 9269 +9309) .

(1.5.25)

s

For example, in an fcc lattice made up of like atoms there are four atoms in a cell

which lie at points 000, 01/2 1/2, 1/201/2, and 1/2 1/20. In this case

S = 1+ exp[in(g, + g3)] + expLin(g, + 93)] + expL[ix(g, +92)]

.

(1.5.26)

If the sum of two indices g, is even, then the corresponding summand in (1.5.26) is equal to (+ 1). If this sum is odd, the summand is (— 1). For instance, for {100}type planes we have g, + g; = 0, which is even, and g, +g, = 1, 9, +93 =1,

which is odd, and therefore, S = 0; ie., there is no reflection. Conversely, for {111}-type planes we have g. + 93 = 9: +92 = 9; +92 = 2, which is an even sum, and S = 4; i.e., reflection takes place.

A simple physical explanation can be provided for this. In the fcc lattice the

{100} planes are reflection planes, and in the case of reflection from two adjacent

parallel cube faces the phase difference is equal to 27. But the atoms that reside in the middle of the faces generate an intermediate plane which produces the phase difference x and thus suppresses the contribution to the scattering amplitude by the adjacent planes. This is schematically illustrated in Fig. 1.28. It stands to reason that such suppression occurs when all four atoms in the cell are alike. In the case of a bcc lattice S = 1 + exp[in(g, + 92 +g3)]. Similarly, the

Fig. 1.28

Explanation of the absence of reflec-

tion from (100)}-type planes in foc lattices

composed of like atoms

1.6

Qualitative Concepts of the Electronic and Nuclear Crystal Structure

43

{100}-planes do not reflect here. However, for a bcc lattice with unlike atoms, for example CsCl, we have F = Fo, + Fo {exp[iz(g, + 92 + 93)]}, and the reflec-

tions from {100} will not be suppressed.

In practice, three methods are used in X-ray diffraction studies:

1. The Bragg method or rotating crystal method. The crystal investigated is rotated about a fixed axis, usually perpendicular to the specified direction of the incident monochromatic (A = const) X-ray beam. With the crystal being rotated through an angle 9, when (1.5.11) is satisfied, the beam is diffracted and the reflected ray is recorded on a photographic plate. 2. The Laue method. In this case the crystal remains fixed with respect to the

X-ray beam. However, the X-ray beam is not monochromatic but “white,” with

the wavelength continuously varying over a sufficiently wide range. Each system of planes, spaced a distance d(g,g29g3) apart, then in a sense picks out from the “white” light the component of wavelength 4 which produces a selective reflection at an angle 9 satisfying (1.5.11). 3. The powder method or Debye-Sherrer method. The sample of fine-grained polycrystalline structure or of compacted fine powder exposed to a monochromatic beam of X-ray light is fixed. In this method the monochromatic beam of wavelength A = const., selects out crystallites in the polycrystal or powder, with planes whose spacing and orientation conform to the condition (1.5.11). At appropriate angles 9 reflections are observed which are recorded as rings on a photographic plate (Debye powder patterns).

1.6 Qualitative Concepts of the Electronic and Nuclear Crystal Structure To gain insight into the atomic structure of close-packed crystals, we recall the experimental fact that the lattice parameter d is close to the mean atomic size 2r,, (Table 1.8). The atoms in a solid “touch” one another just as spheres do. At the same time we should remember that the electronic shell of an atom is a highly complex entity. Roughly, the electronic shell may be thought of as composed of subshells, which are specified by two quantum numbers: the main quantum number n = 1, 2, 3,... and the orbital quantum number / = 1, 2,..., (n— 1);

21+ 1 electrons in a shell with a given n and / are numbered by magnetic

quantum

with

numbers

—! < m < |. In addition to this, there may be two electrons

these three quantum

numbers—n,

|, and

m—in

keeping

with the two

possible spin orientations (the spin quantum number o = + 1/2). Thus the number of electrons in a shell with quantum numbers n and I is equal to 2 (2/ + 1). For states with different orbital numbers the following notation has been adopted: s(/I = 0), p(/ = 1), d(l = 2), f (1 = 3), etc. The filling of a shell, i.e., its

electronic configuration, is designated as ni", e.g., 1s?, 3d°, etc.

44

1. Introduction. General Properties of the Solid State of Matter

Table 1.9. Successive filling of electron shells Configuration with given n and | n l=s p d Sf

1

Is?

3

3s?

2

2s?

k

— Symbol of shell

2

3p® = 3d?°

4p®

4d'°

4s

6

6s?

6p®

6d'°

6 f!*

5s?

Total number of electrons in shell

Sp®—

d!°

Is? = 7pS

7d!°

K

8

4s?

7

h

2p®

4

5

g

S14 Sgit

7h!

L

18

6g!®

7g®

M

32

6h?

Th?

7k26

N

50

0

98

Q

P

A successive filling of shells with electrons is shown in Table 1.9. However, in

reality, the filling follows a somewhat different scheme. Sometimes it appears more advantageous to start completing a shell with a larger value of the main quantum

number n but with a smaller value of the orbital |. For example, after

'8Ar of 1s?2s?2p°3s?3p® configuration it would be reasonable to start com-

pleting the 3d shell with its ten sites, but the configurations of the next two

periodic table elements actually are '°K (1s?2p°3s?3p®...4s) and ?°Ca (1s?2s?2p®3s?3p® .. . 4s?). It is only from scandium ?'Sc that the unfilled 3d shell starts to fill. This tardy completion encompasses seven more elements

from ??Ti to ?®Ni and terminates at copper 7®Cu whose atomic shell is filled

up

“correctly”

(according

to

the

scheme

presented

in

Table

1.9:

1522s? 2p® 3s? 3p®3d'°4s). Then the completion of the 4s and 4p shells goes on

without disturbance up to krypton °°Kr, but again there is a blank for rubidium >7Rb and strontium >°Sr: the 5s shell fills, whereas the 4d and 4f shells remain

empty. The 4d shell starts to fill from yttrium 3°Y to palladium *°Pd, and the

filling of the 4f shell commences with cerium *®Ce and terminates with ytterbium 7°Yb. Table 1.10 presents equilibrium atomic configurations for all the 92 elements of the periodic table. Elements said to be normal are those in which all shells, except perhaps for the outermost shell, are successively filled from the beginning to the end or in which there are completely vacant shells. Transition elements are those in which Table 1.10. Equilibrium electron shell configurations for the elements of the periodic table

Is

'H

2He

1

2

SLi

“Be

1.6

Qualitative Concepts of the Electronic and Nuclear Crystal Structure

Table 1.10 (continued)

[_|™Na]"Mg]"al] “si | =P] ie tana 2.) 2 | 2 | 3p tesa [os [19K [29Ca | ase | 2271 | 28v | 3d 1 (eels. eibetli2,| 2 | 2 | 2 2°Zn | 24Ga |32Ge | As | 24Se | ™ meant 2 lia | 2

| 4p

Pie

|

Ss | wep

*s | cl] (2 [2 | Toa fos 2cr |?8Mn| rs: lcs [1 | 2 *Br | 3¢Kr for) 2

el

Ag 2 Pd 2Fe |2°Co | 2*Ni | °Cu [oe | | & [10 [2[2 [2] 1

6

|

J22rb] 2sr | agy | *°zr |*Nb [Mo] Te [Ru |*9Rh | “Pd

|

$5Cs | 56Ba

4d .

fl ep Se le) it abl Ee 47g] “8ca | “In | 5°sn [5'sb ]52Te | 951 | Meee 2-| 2] 2 )2 ) 2/2 4] Sp lisa arf se] |

ae a 54Xe 2 6

ale | a0

[37La | 5*Ce | 5°Pr |6°Nd |*'Pm | Sm

4f dice = | oS ING Sd j fa cepee {2 |2{[2{[2{2)2 [2 io ®3Eu| Gd | ©Tb | “Dy |°7Ho | Ez |6°Tm| 7°Yb af| 7] 7 [8 [10 | 1 | 12 | 13 | 14 Sd qe) 4 mere > | oa] 2 [2 | 22]. |

ALu|

Hf | Ta

|W

|75Re | 7°Os | 77Ir | "Pt | 7?Au

|°°Th

|°'Pa | °?U

merry 2 |3 [4f[s5 [7 [7] Memmeuto2 |2|2 {12 [1 [2 srg] *T1 |*Pb | Bi [*Po | At |*Rn meee | 2 | 2 | 2 | 2] 2 | 2 6p yo aS Joa [ls] %6 87Fr | *Ra

|fAc

sf aS 6d tea | pes Smetenhiy2;.|.2 | 2 | 2 | 2

9 | 10 ]1 [1

45

46

1. Introduction. General Properties of the Solid State of Matter

the previously unfilled shells are filled up. Of the 92 elements (up to the transuranium elements—Table 1.10), there are 40 transition elements. They are divided into three groups: 24 d elements that constitute three subgroups of eight elements with an incomplete 3d sheil (iron group), 4d shell (palladium group), and 5d shell (platinum group); 12 felements with incomplete 4/ shells (rare-earth lanthanide group); and four mixed d-f elements with incomplete 6d and 5fshells (actinide group).

All transition-metal crystals are metals. Figure 1.29 presents radial electron densities P?(r) = 4nn(r)r? in different shells of the Gd* ion, calculated according to the Hartree-Fock method [1.21]. It also shows the magnitude of half the lattice parameter. As seen from the figure, only the outer 6s shells of the adjacent atoms will superpose and undergo substantial deformation as compared with the shell of an isolated atom. An individual wave function y,(r) may be approximately introduced for each electron in the atomic shell (the exact many-

electron wave function depends on the coordinates and spins of all atomic-shell electrons). The parameter characterizing the effect of the interatomic interaction in the crystal on the motion of an electron, described by the wave function y,(r), may therefore be taken to be the product of the wave functions of two electrons of adjacent atoms

(1.6.1)

+ na) , S.(r)= Pr (r)Wa(r

Fig. 1.29. Radial charge

densities for 4/,

5s, and 6s electrons in a Gd*

ion as a

function of the distance from the nuclear

center

with n = 1 for the nearest neighbors and with a = a’ for the same states. If the function S“(r) describing the overlap of the wave functions of the neighboring atoms is small in the entire space, the interaction between the corresponding electrons will be weak and, in the crystal, they will move in nearly the same way as in isolated atoms. If this function is different from zero at least in some regions

1.6

Qualitative Concepts of the Electronic and Nuclear Crystal Structure

47

of space, the interaction is liable to be large enough and the electron motion may

alter substantially in character. The wave functions of outer- (valence-) shell

electrons overlap more strongly than those of inner-shell electrons. For each shell, a mean effective radius r, may be introduced which corresponds to the maximum of its radial density. Then the dimensionless parameter

¢ = 2r,/d

(1.6.2)

for electrons that practically do not change the character of their motion during condensation from a gas into a crystal will be ) atom (ion)

|Jmn'>=|m)>|n’> ,

(1.7.9)

where m and n’ are the respective many-electron states of the shells of the first and second atom (ion). The atomic (ionic) nuclei, in virtue of their large mass, are treated classically as electric field sources (adiabatic approximation—Sect. 1.9). Specifically, the wave function (1.7.9) of the ground state of the system involved

is |00’> =|0)|0’>, ie., the product of the ground-state wave functions of each of

the atoms (ions). Strictly speaking, the choice of (1.7.9) is not altogether correct,

since, even in the absence of a dynamic interaction between two-atom (ion) shell electrons,

it is necessary

to take

into account,

in accordance

with

the

Pauli

principle, their statistical interaction, i.c., the antisymmetry of the wave functions with respect to the permutation of electrons (more exactly, the permutation of the spatial and spin coordinates of every single electron pair) belonging

to different atoms (ions) (the wave functions of each individual ion may be assumed to be correct in this sense). However, it may be shown (more details will

be given later) that such antisymmetrization leads only to the occurrence, in the interaction energy, of contributions that are exponentially small for R— oo. According to conventional quantum-mechanical perturbation theory, the first-order correction for perturbation (1.7.8) to the ground-state energy is equal to the diagonal matrix element of the operator (1.7.8) which in this case has the form

E,(R) = (00'|Ajq:|00'> = aq‘/R .

(1.7.10)

In fact, since the electron-density distribution in the closed electron shell of

atoms (ions) is spherically symmetric, not only the dipole moments are absent—

= ¢0'|d'|0’) =0

(1.7.11)

but also the quantity

= COIS, e(L3(r,-m)? — 1? J10>

(1.7.12)

becomes identically equal to zero. A quadrupole moment tensor is introduced here:

— 17 Sap) Onp = >. 6(3%iahip

(1.7.13)

52

1. Introduction. General Properties of the Solid State of Matter

We

wish to prove the validity of (1.7.12). In virtue of the aforementioned

spherical symmetry of the shells of isolated atoms (ions) with closed shells, the quantities (1.7.12) should not depend on the direction of the vector m. Averaging

(1.7.12) over spherical angles then yields a zero value. Indeed, for any vector a we have

2n

«

fdw(3(a@-n)? —a?] = J dp J sin 9d9(3acos? 9 — a?)

(1.7.14)

1

=na?

§ dx(3x?—1)=0,

-1

where the polar axis is aligned with the vector a and the substitution cos $ = x is used. Since (1.7.12) does not depend on the vector a, it is equal to its mean value,

ie., zero, as follows from (1.7.14). This is the case provided

(1.7.15)

.

910>=0 = ¢0'|Qzg|0'> €01Q.

Allowing for (1.7.11) and (1.7.15), we obtain the result (1.7.10) for the diagonal matrix element of the operator (1.7.8). Moreover, this result remains valid also

when we take into account the arbitrary-order terms in the small parameter ro/R in (1.7.8). To see this, we calculate the electric potential p(R) produced by a nonpointlike ion with spherically symmetric distribution of the electron density Q(r) in its shell. Let the ion be circumscribed by a sphere of radius R with its center coinciding with the center of the nucleus of the ion. Then, the electric field intensity E(R)= —(0@(R)/OR)n is oriented along the vector R and its flux through the above sphere is equal to —42R?0@(R)/OR. According to the Gauss—Ostrogradskii

theorem, we now

—4nR?0e(R)/dR =4nq(R)

have

,

(1.7.16)

where

(1.7.17)

q(R)= f 4nr? o(r)dr 0

is the charge contained within the sphere of radius R. Substituting (1.7.17) into (1.7.16) and performing integration by parts, we find

@(R)= | drr-2q(r)= — f d(d/nyacr) R

x

R

xe

=q(R)/R—§ drr~' dq(r)/dr = q(R)/R—4n { drro(r) . R

R

(1.7.18)

1.7

Fundamental Concepts of the Chemical Bonding in Solids

53

From (1.7.18) it follows that if g(r) decays exponentially with r— oo, as is the case for free atoms and ions, the correction to the asymptotic value of the

potential q(co)/R is exponentially small, too. Thus the quantity E,(R) is nothing but the electrostatic-interaction energy of nonpointlike ions that are in the

ground state. Expression (1.7.10) therefore holds to within exponentially small terms. When taking account of the latter, however, in the zero approximation we must allow for the antisymmetry of the zero-approximation wave functions with

respect to the permutation of the electrons of two ions. A quantum-mechanical

calculation [1.23] thus leads to

AE(R)= A(R)exp(—7R) ,

(1.7.19)

where y is a constant for a given pair of atoms (ions) and A(R) is a function R

which is smooth compared with the exponent and which also is normally replaced by a constant. An interaction of the type (1.7.19) is referred to as the

Born—Mayer repulsion. It arises from both the purely classical effect of the electrostatic interaction of nonpointlike atoms (ions) and the Pauli principle, as well as from the overlap of the wave functions of different atoms (ions).

Now we proceed to a calculation of the second-order correction of perturba-

tion theory respect to H,,, from (1.7.8). For this correction, according to the general theory, we have

E,(R)

a

{00'| HayEES’ ”n’ >|? ETE.

(1.7.20)

where E, and E,,. are the energies of the corresponding states of noninteracting atoms (ions). In virtue of the orthogonality of the excited states to the ground

state,

= +6,,||? , (1.7.23)

>]? > dg_1. Similar to Ge, which is situated between Ga and As in the same

row, this compound

possesses a diamond lattice with saturated two-

electron bonds (Fig. 1.35). For four neighboring atoms we again have eight p

valence electrons of the shells 4s?4p and 4s?4p°. Of these, the larger share (5/8) consists of As atoms (BY), and the smaller share (3/8) of Ga atoms (A"),

Therefore, the shaded covalent-bond bridges in Fig. 1.35 are not symmetric, as

compared to those in Fig. 1.34, but are thick near As ions with a core charge +5e and thin near Ga ions where the charge of the core is equal to +3e.

Fig. 1.35. Electronic structure of A™BY (GaAs) compounds with

hybrid covalent and ionic bonds (diamond-type lattice in a plane projection)

16

1, Introduction. General Properties of the Solid State of Matter

Gallium arsenide is a typical semiconductor of the class A''BY and features a hybrid bond in which the covalent bond prevails. Consider a binary compound that consists of elements contained in the

second and sixth columns A"BY'—for example, ZnSe with the respective valence shell configurations 4s? and 4s?4p*, and a diamond lattice. Again, two-electron

bridges (Fig. 1.36) arise between the nearest neighbors. The asymmetry of these

bridges is more pronounced than that in the case of GaAs, because the ionic core of zinc has a charge + 2e, and the core of selenium + 6e.

Fig. 1.36. Electrpnic structure of A"BY! (ZnSe) compounds with

hybrid covalent and ionic bonds (diamond-type lattice in a plane projection)

Finally, in A'*BY"~ compounds such as Cu* Br~ we are dealing with a typical ionic bond. There is no diamond lattice in this case. Nor are there any traces of a homeopolar bond in these compounds. As can be seen from the above

examples,

the drastic differences between

covalent and ionic bonds

that have

been adopted for their classification are, in many cases (e.g., in A''BY and A"BY! compounds), of a relative nature only.

1.8.3 Metals, Their Alloys, and Compounds In the solid state, about 70 of the 92 stable elements are metals, accounting for

nearly 75% of all the periodic table elements. In the case of normal elements, all the elements of the first and second columns are metals, i.e., the alkali metals and

the alkaline-earth metals, respectively. In the third column all the elements,

except

for

boron,

are

metals.

The

fourth

column

contains

only

one

metal,

namely, lead. Tin and carbon are metals only when in certain modifications, and

Si and Ge are semiconductors. The normal elements of columns V through VIII

are all nonmetals, and the transition elements are all metals, including the noble

metals Cu, Ag and Au and the elements Zn, Cd, and Hg. The most remarkable of the physical properties of metals is their high electrical and thermal conductivity. As the temperature is lowered, the specific electrical

resistivity

of metals

falls off, tending

to a minimum

value

when

1.8

Types of Crystalline Solids

77

T — 0 K. This leads to the assumption that the valence electrons of a metal form

a highly mobile system of conduction electrons, or an “electronic liquid”, which

is easy to accelerate in an electric field. The nature of the metallic bond, as well as

of the homopolar bond, could not be understood in terms of classical physics (that was another of its “catastrophes”), and only quantum physics has introduced theoretical enlightenment. Energetically, the metallic bond is weaker than

the ionic and covalent bonds. This is particularly true of normal metals. In these,

the bonding energy at room temperature ranges from 15 kcal/mol (= 62.5 J/kmol) (Hg) to 92 kcal/mol (~ 386 J/kmol) (Au). In transition metals the bonding energy is higher. For example, for W it is equal to 210 kcal/mol ( 875 J/kmol), which is due to the contribution that the electrons of the closed d and f shells make to the bonding energies. At least in part, this bond is of a covalent nature.

The difference between the metallic bond and the covalent bond

arises

primarily from the magnitude of the electron-ion interaction. For typical representatives of metallic-bonded crystals, i.e., alkali metals, this interaction is relatively weak. Therefore, as a first approximation, it may be assumed that the metal lattice is stabilized by the energy of the free-electron gas, and the role of the ionic cores is, chiefly, to assure electroneutrality (see the “plasma model of the metal”, Sect. 5.1). The energy of typical metals consists of a large structureindependent contribution and a relatively small structure-dependent contribution. The energy difference for the various structures with the same mean electron-ion density therefore is not large and that is why it is very typical of metals to exhibit polymorphy. Thus, lithium and sodium exist at atmospheric pressure in low-temperature hcp and high-temperature bcc phases. Typically, metals possess very closely packed bcc, fcc, and hcp lattices. As the charge of the ionic core (i.e., the valence) builds up, the electron-ion interaction strengthens, and the electron-density distribution becomes nonuniform. A tendency appears for piling up charges on the bonds, and we proceed to the case of covalent

crystals. For polyvalent metals, the bond is normally of an intermediate type;

structures of very low symmetry occur quite often in which the number of nearest neighbors is small (for example, three nearest neighbors in the lattices of gallium and bismuth). Very complicated types of structure occur in transition metals such as manganese, tungsten, etc. For a more detailed treatment of the

chemical bonding in metals, see [1.31].

Chemical bonds in alloys and intermetallic compounds are even more varied. Here we deal with a multitude of behavior types—from the practically

complete insolubility of one type of metal in other types to the formation of a continuous series of solid solutions. The problem of alloy formation is an extremely complicated one and so far we have to confine ourselves to mainly

empirical and semiempirical rules [1.32]. We just note here that in the case of alloys or compounds of metals whose valence differs substantially one may talk of a bond

that is intermediate between

the metallic and ionic bonds. Thus, a

plurality of intermediate forms exist between the three types of “tight” chemical

78

bonds

1. Introduction. General Properties of the Solid State of Matter

(ionic, covalent,

and

metallic), of which

electron

transfer between

the

structural units of the crystal is typical. Now we proceed to a consideration of the type of solids with weak chemical bonds.

1.8.4 Molecular Crystals In molecular crystals the bonding forces within the molecules located at the lattice sites are appreciably stronger than the crystalline binding forces between

the molecules. Since the electron shells of the molecules are, as a rule, closed, the

intermolecular forces in the crystal are due to polarization (van der Waals forces). Note that neutral atoms or molecules polarize not only in the presence of a charged particle. Polarization also occurs for two neighboring neutral and isotropic

particles.

The

potential

energy

of these

van

der

Waals

forces

is

inversely proportional to the sixth power of the interparticle distance (r~°) for sufficiently far-off particles, when r > 2r,,. Since this condition is not satisfied in crystals, the calculations for them are highly complicated. Van der Waals forces

are not only essential in molecular crystals—they introduce a small correction into the energy of ionic crystals, and also into the energy of metals. Strictly

speaking, they should not be regarded as purely pairwise interaction forces, because the polarization of a pair will be affected by other neighbors as well. Roughly, however, they may short-range forces.

be viewed

as pairwise-interaction, central, and

Typical representatives of molecular crystals are the normal elements of the eighth column of the periodic table—rare gases; molecular hydrogen H,; oxygen O,; nitrogen N,.; the halides Cl,, I,, etc.; molecular NH;, CO,, CH,, and a huge number of organic compounds, including the most complicated

biological systems. As a rule, the structure of these crystals is determined by the requirement of a close-packed arrangement of molecules. Since the latter

normally have a complicated form, the structure of the crystal turns out to be complicated too [1.33]. The bonding forces in molecular crystals are hundreds

of times smaller than those in ionic crystals. For example, in helium they amount to 0.053 kcal/mol (= 0.218 J/kmol); for argon they are as small as 1.77 kcal/mol (¥ 7.37 J/kmol); and for CH,—2.4 kcal/mol (= 10 J/kmol).

1.8.5 Hydrogen-Bonded Crystals Hydrogen-bonded crystals are also molecular-type crystals. However, the bond in them differs considerably from both the ionic type of bonding and van der

Waals bonding, and is characterized by a shared proton (ion of the hydrogen

atom). To some extent this type of bond may be thought of as being of an ionic

1.8

Types of Crystalline Solids

719

nature—hydrogen transfers its electron to a neighboring atom and thus makes it a negative ion. The small size of the proton reduces the number of its nearest neighbors to a minimum, i.e., two neighbors. The bonding energy in these crystals is as low as 5 kcal/mol (~ 20.5 J/kmol). Hydrogen bonds and van der

Waals bonding are vitally important to biology in the formation of the structure

of proteins and nucleic acids [1.34].

The most common hydrogen bonds are OH... O and NH... O. Very frequently the bond turns out to have two minima, i.e., the proton involved in the bonding may be in two equivalent or almost equivalent energy positions (i.e., a two-well potential for the proton [1.35]). Proton ordering on bonds of this type is one of the mechanisms of structural phase transitions in solids [1.36].

1.8.6 Quasi—One-Dimensional and Quasi-Two-Dimensional Crystals

A very interesting class of substances is represented by long polymeric mole-

cules. Strong covalent bonds exist between the constituent atoms, whereas the

interaction of different molecules arises from relatively weak van der Waals or hydrogen-bonding forces. If there are conjugated bonds in the molecule, electron transfer along the chain is also possible. Thus, according to the conductivity type, the (SN), system (Fig. 1.37) is metallic, as are a number of organic

polymers, although semiconductive behavior in such systems is actually much more common—the result of a phenomenon treated in Sect. 4.4.2. Of the pure elements, selenium and tellurium form “chain” structures with relatively weak bonds between the chains. In connection with some biological problems, studying electron motion in quasi—one-dimensional systems is particularly interesting because such chains are a constituent part of many biologically active molecules (chlorophyll, vitamin A, etc.).

=

7

Fig. 1.37. Inorganic polymer (SN),

Not only because of their different electronic properties do we distinguish this type of solid from molecular crystals. In a classification according to chemical bond types, crystalline polymers also should come under a special heading, since the chain structure depends on the strength of the covalent bond, and the packing of the chains is determined by van der Waals bonding or hydrogen bonds.

Now we briefly examine quasi-two-dimensional crystals. A number of substances possess a layered (or quasi-two-dimensional) structure. Examples include graphite, boron nitride, and compounds with a formula of the MX, type

(where

M

is a transition metal of groups

IV-VII, X =S,

Se, Te). As an

1. Introduction. General Properties of the Solid State of Matter

t!

80

Fig. 1.38. The graphite lattice

illustration, Fig. 1.38 presents the structure of graphite. In such substances the chemical bond of the atoms is much stronger in the layers than between the layers. The interlayer spacing may be made larger (and the bond weaker) by

introducing foreign molecules into the space between the layers (intercalation).

Low-dimensional systems are highly specific in their lattice and electronic

properties. Normally, the interelectron interaction in these systems plays a more important part, with electronic phase transitions often occurring (e.g., metalsemiconductor transitions).

1.8.7 Quantum Crystals In Sect. 1.2 we mentioned the quantum “zero-point” oscillations of atoms (ions) in solids with a mean amplitude 0),

which are depicted in Fig. 2.9 for some direction g. These branches give a linear

108

2. Dynamic Properties of the Crystal Lattice

i)

Fig. 2.9. Dependence of the frequency w of a monatomic threedimensional lattice (for some direction in wave-vector g space) for longitudinal L and transverse 7,, 7, oscillations within one Brillouin zone

9

dependence of w, on q in the vicinity of g = 0, the slope of the curves being equal to the velocity of propagation of the corresponding oscillation. We wish to elaborate on this important problem. The potential energy (2.1.28) should not vary during shifting of the entire crystal as a whole, i.e, when uj, = u/ = const. We leave as an exercise for the reader the proof that this fact entails the identity }.,,. G,.(¢ = 0) = 0. It thus follows from (2.1.37) that with q 0, w,—0 and three s’-independent solutions exist for j = x, y, z. The existence of acoustic branches for which w, = 0 at q = 0 thus comes from the

fact that the lattice is translationally invariant. This is a particular case of Goldstone’s fundamental theorem [2.4], which relates the symmetry properties of a system to the existence of low-frequency modes in it and has numerous

applications in the theory of magnetism, superconductivity, and phase transitions. As |q| increases and approaches the boundary of the zone, the linearity is disturbed;

at the boundary

itself, by contrast

with

the one-dimensional

case

(Fig. 2.2), the point of contact with the horizontal tangent may not be reached.

However, the w, curves in the general case remain periodic, the period being

related to the reciprocal-lattice vectors; namely, the points g and g + 5* (with b* being a reciprocal-lattice vector) are equivalent (i.e., possess equal values: Wy = Weise):

For a diatomic ionic crystal with o = 2, the equation for «? is of sixth degree,

and in the general case we obtain six spectral branches (0, i= 1,2,3,...,6), of

which three are acoustic and three are optical. This is diagrammatically represented (for some direction of g) in Fig. 2.10. The extended-zone diagram in

3

2 {

Fig. 2.10.

Dispersion

relation

branches for a diatomic lattice

of acoustic

(1-3)

and

optical

(4-6)

2.1.

The Dynamics of the Ionic Lattice

109

Fig. 2.11. Surfaces of the frequency constant

cw! = const in wave-vector g space for one of

Vi

the oscillation branches in the extendedzones scheme

Fig. 2.11 shows (for some definite cross sections in g space) the picture of periodic constant-frequency surfaces for one of the spectral branches {). In the general case, the number of spectral branches will be equal to 3c: (s=1,2,...,0).

(2.1.38)

2.1.4 Quantization of Ionic-Lattice Vibrations Prior to considering the physical properties of ionic lattices, we wish to give a quantum description of their vibrations. As already stated in Sect. 1.9, an efficient way of solving many-particle problems is by using the unitary transformation technique, which allows the interaction term in the Hamiltonian to be brought into diagonal form so that the complicated motion of the system can be described, approximately, as the motion of an ideal quasiparticle gas. This is what we actually did when considering the classical problem of the oscillations of arrays and a three-dimensional crystal. In fact, the solution of (2.1.3 or 6) is nothing but a unitary transformation, which diagonalizes the oscillation array. In this classical problem an elementary excitation is a sine wave. Let us see how this situation changes in the quantum case. We add to the generalized-coordinate operators 1, the , conjugate-momentum operators p,. Then the kinetic-energy operator will be T= ¥, 6? /2m, where the summation is taken over all N array sites, and the potential-energy operator in the nearestneighbor approximation is given by (2.1.1). The Hamiltonian of the system. #, which is equal to the sum of the operators T and U, will be H=T+U=

2 (8? /2m + ati?) — 5

=P p7/2m + ati? — aD tty. U

thea + tity

,

a)

(2.1.39)

110

2. Dynamic Properties of the Crystal Lattice

with the operators 4, and p, satisfying the commutation relations [4,,p)J]-=ihdy

,

[ud]

= ([p, p,-J- =0.

(2.1.40)

The symbol [4, 6]. = 4b — bd stands for the commutator. The first sum of the right-hand side of (2.1.40) is the energy operator of N independent harmonic oscillators, and the second sum takes into account the interaction of nearest neighbors in the array. The unitary transformation diagonalizing (2.1.39) has the form ~

i,=N UU?

[ G.cosat -

1

ine, P,singl | Z

q

B= N7"2¥ [mo,U,singl + P,cosql!]

;

q

(2.1.41)

where w, is defined by (2.1.5), and the operators U, and P, satisfy, by virtue of (2.1.40, 41), the commutation relations

[U,, Py]- =ih6,,,

(U,,Uy]- =(P,, Py]- =0,

(2.1.42)

the proof of which is left as an exercise for the reader. Using (2.1.41, 42), the Hamiltonian (2.1.39) may be shown to assume the diagonal form in the new variables

30" = YB? /2m + (mw? /2)0,7) «

(2.1.43)

We also leave as an exercise for the reader the proof that this is so. The operator

(2.1.43) is equal to the sum of the Hamiltonians of linear harmonic oscillators

with frequencies w,. The energy of the system is equal to the sum of the energies of such oscillators

€=>(n,+)ho,, q

mn, =0,1,2,....

(2.1.44)

Thus the oscillation of the array is represented as a “gas” of independent effective oscillators, that is, quantized sonic waves with frequencies w,. A

quantized sonic wave with a wave vector q, frequency w,, and energy (n, + 1/2)ha, may be regarded as a set of n, quanta with an energy hw, each and

with a zero-point oscillation energy hw,/2. These quanta are referred to as phonons. Sometimes, not altogether correctly, they are said to be quantized sonic waves. In fact, a sonic wave is associated with n, phonons plus the zero-

point vibrational energy. For this reason the phonon should be called the sound quantum. The quantity hw, is the smallest portion of energy above the ground-

state level hw,/2. The phonon, therefore, is the collective excitation of the ionic lattice as a whole.

2.2.

The Specific-Heat Capacity of the Lattice

11

One more transformation may be carried out on the Hamiltonian (2.1.43):

5, = (2mho,)-!? P, — i(ma,/2h)"? U, , f= (2mhw,)~'? P, — i(me,/2h)'? U, .

(2.1.45) (2.1.46)

It can be readily verified, by means of (2.1.42), that the operators b, and bt obey

the commutation rules

[By Bo 1- = 549

(bs, btJ- = [b,, by J- =0

(2.1.47)

and the Hamiltonian (2.1.43) will take a more compact form

= ¥ (bf b, + 1/2)ha, q

=4Y ho, + bf bho, = 8) + Yi,ho, . q

q

q

(2.1.48)

Comparison of (2.1.48) with (2.1.44) shows that the operator bt b, is a quasiparticle number operator; i.e. 7, is a phonon number operator. The operators be and b, are called the Bose operators, and phonons are Bose particles, or “posons, ‘similar to light quanta, or photons. The wave function of the system of bosons is symmetric with respect to permutations of the coordinates of any pair of particles. In a statistical description of bosons the Bose-Einstein statistics are used (hence the name of these particles). Specifically, the mean with respect to the number of bosons 4, is given by the well-known formula (Sect. 2.7.2): A, = [exp(hw,/kgT)—1]"'

,

(2.1.49)

and the mean energy of the oscillator is equal to &, = (fi, + 1/2)ho, .

(2.1.50)

In the three-dimensional case it is necessary to replace q by the vector q, and also to take into account the polarization for the acoustic and optical branches.

2.2 The Specific-Heat Capacity of the Lattice As far back as 1819 Dulong and Petit discovered an empirical law according to

which the atomic heat capacity in the case of a constant volume is practically

112

2. Dynamic Properties of the Crystal Lattice

constant (independent of 7) for an overwhelming majority of solids and is equal to

Cy = 6kcaldeg™! mol™'! ~25Jmol"'K™!

.

(2.2.1)

This is easy to explain using the general laws of classical statistical mechanics and the concepts about the oscillatory nature of the thermal motion of crystals. A mole of a monatomic crystal contain N, atoms (N, is Avogadro’s number ~ 6.02 x 1073 mol~'). Therefore, the number of degrees of freedom is equal to 3N,. Acording to the law of equipartition of energy, each degree of freedom has on the average k, 7/2 of kinetic energy. If we visualize the crystal as an assembly of 3N, harmonic oscillators, their total energy is equal to the sum of the kinetic é, and the potential e, energy. It is also known from statistics that the mean

values of these energies for a harmonic oscillator are equal to each other and

each of them is equal to 3k, 7/2, and their sum is 3k,T. (We leave it as an exercise

for the reader to prove all this.) Therefore, the thermal energy of a mole of a monatomic crystal is equal to & = 3N,kgT= 3RT, where the gas constant R ~ 2 kcal/mol = 8.31 Jmol”! K~', and for the heat capacity we have

c= (#)

v

=3R = 6 kcaldeg™' mol”! = 25Jkmol7'K7=!

.

(2.2.2)

Thus the theoretical formula (2.2.2) confirms Dulong’s and Petit’s experimental law (2.2.1). Originally, the experiments of Dulong and Petit related to a comparatively narrow temperature interval (close to room temperature, ~ 20°C). As that interval was extended, two tendencies were noticed: the quantity C, increased with increasing 7, as the melting point was approached (Sect. 2.3.2), and Cy decreased during cooling, in which case a general law was revealed—C, > 0 when

T — 0 (Fig. 2.12). After Nernst discovered in 1906 the thermal theorem

(third law of thermodynamics), that property of the heat capacity proved to be Cy, kcal, /deg mole

0

0s

1.0

15

Tidy

Fig. 2.12. Comparison of experimental data for the molecular heat capacity C, (in kcal/deg mole) of some solids as a function of reduced temperature 7/9p with the Debye theoretical curve (2.2.16)

2.2.

The Specific-Heat Capacity of the Lattice

113

an immediate corollary of the thermal theorem. In 1906 Einstein was the first to explain qualitatively the curve (Fig. 2.12) for C,(7), using quantum theory. He suggested that the crystal be treated as a totality of 3N, linear quantum oscillators with equal frequency w, the energy of which is always a multiple of the minimum quantity hw: nhw (n = 1,2, . . .);i.e., there is a spectrum of discrete nondegenerate states (g, = 1).

Using the standard formula for the partition function Z, we obtain

Z(T)=

x exp[—(n+ 1/2)hw/k,T]

(2.23)

= ¥ [exp(—hw/kyT)]"*" a=0

_

_exp(—hw/2k,T)

~ L—exp(—fha/kgT) By definition, the mean energy of a linear oscillator is equal to hw é=>+ Zi)1 mm2 nhwexp(—nhw/k,T)

ho

0

=F ~ aarecry 9 EeerC—ntoita? | é = Hiker)

82)

ee

hay + 7:

Employing (2.2.3) for Z(T), we find

._

&=

hw 7

ho

=

1

a/keT) +

— exp(—hw/kgT)J

hw

* aplio/kT)—1°

(2.2.5)

Equation (2.2.5) yields z

3N,hw

&= apliol,T)ci t

3N,hw

(2.2.6)

for the molar energy and

_ (9%) =_

Cy = (3),

3N,kp(hw/kgT)

exp(lica/kT)— 2 [exp(hio/kpT)

1]?

(2.2.7)

114

2. Dynamic Properties of the Crystal Lattice

This is the celebrated Einstein formula for the atomic heat capacity of a crystal. Let us consider its asymptote at high and low T. At high temperatures kgT> hw, or hw/kgT 1, the unity in the denominator of the fraction in (2.2.7) may be neglected in comparison with exp(hw/k, 7). Then the approximate result for C, will be Cy © 3Nykg(hw/kgT)? exp(—hw/kgT)

.

(2.2.8)

In the limit T—0 the exponential factor tends to zero faster than the power

factor (hw/k, 7)? increases, and therefore, by (2.2.8), C, +0 when T—0 K

as

exp(—fhw/k,T), which agrees with the Nernst theorem and is qualitatively consistent with experiment. However, the experimental curve in the vicinity of 0 K obeys the power law (~ T°) rather than an exponential law. This difference apparently is due to Einstein’s assumption of the value of w being the same for all atomic oscillators, which is too crude an approximation. A further refinement of the theory of the heat capacity of solids was made by Debye [2.3]. He took account of the fact that the crystal had an oscillation frequency spectrum (2.1.37). Confining himself to the continuum approximation, he used only one acoustic branch, assuming the optical branches to be absent and

the three acoustic branches

to coincide.

Furthermore,

he assumed

the

dispersion relation to be linear. To take into account the discreteness of the crystal and the correct number of degrees of freedom , Debye, in his formulas for the thermal energy of the crystal, extended the integration not over the Brillouin zone but over a sphere in q space of finite radius q,,,, (thus cutting the spectrum off at a minimum wavelength 1,1, = 27/qmax)- Taking account of the fact that the density of allowed values of g in g space is equal to V/(2z)*, with V being the volume of a solid [(2.1.13) for a one-dimensional array], the value of q,,,, can be determined from the equation

(41/3) Qiuax (V/82°) = N

OF

max = (627 N/V)? = (6n7/v,)'? ,

(2.2.9)

where v, is the volume per lattice site, which may be represented as a sphere of equal volume of radius r, (Wigner-Seitz sphere): ve =$ ard,

dunax= (92 /2)"3 /r,

and

Amin = 214max® 2, OF, -

(2.2.10)

2.2.

The Specific-Heat Capacity of the Lattice

115

Then the maximum frequency according to Debye is max=

Vs Imax

-

(2.2.11)

In addition, since the Debye model involves not one frequency, as in the case

of the Einstein model, a simple multiplication by 3N, does not permit us to

proceed from (2.2.5 to 6), so we have to integrate over the Debye sphere: omx

hw D(w)dw

(2.2.12)

€= |} exptho/kyT)—I

Calculating (2.2.12) requires a knowledge of the spectral density D(w) The quantity D(w)dw is equal to the product of 3N, by the ratio of the volume of the spherical layer 4q?/dq to the volume of the entire Debye sphere

(41/3) dina: ie.,

D(w) dw = 3N, 4nq7dq/(4n/3)q2,..=9N, 07 dw/w3,, .

(2.2.13)

Here we have employed (2.2.11) and the linear dispersion relation w = v,q. As

seen from (2.2.13), the spectral density according to Debye has the form of a parabolic law D(w) = w? (Fig. 2.13). Substitution of (2.2.13) into (2.2.12) yields

=

ONgh w3,.,

" w> dw 9 exp(hw/kgT)—1’

Dw)

D

Fig. 2.13. Spectral density after Debye max

and for the heat capacity we then have

C= (*) _ 9Naky ** (ho/ky T)? 0? exp(heo/ky T)do aT ymax 0 ——«(Lexp(Fio/kgT)— 1)" =

@max

2

42

(2.2.14)

Introduce a new variable x = hw/k, T and determine the Debye temperature

9p = hOmax/kp -

(2.2.15)

116

2. Dynamic Properties of the Crystal Lattice

Ultimately, we find for C,

Cy =9R(T/9p)>_

9)/T

f (eX— 1)? x4e% dx .

0

This is the celebrated

Debye

heat-capacity formula.

(2.2.16) At high temperatures

(T > 9p) the upper limit of the integral in (2.2.16) is very small (9p/T < 1), and

therefore the integrand may be expanded in a power series of x. To a first approximation, this gives 9,/T J (x*/x?)dx =4(9)/T) oO

.

Then it is simple to see that (2.2.16) assumes the form of Cy ~ 3R; i.e., we come

again to the Dulong and Petit law. At low temperatures the upper limit of the integral in (2.2.16) may be replaced as 9)/T — oo, and we arrive at the tabulated integral j x*e*(e*— 1)°2 dx = 4n*/15

0

,

and then find for C,

Cy & (12n4/5)R(T/9p)° ;

(2.2.17)

ie., the “Debye 7° law”. Figure 2.12 shows that it gives a very good fit to experimental curves in the region of low temperatures. The temperature 9p is determined from experimental data and tabulated. It will suffice here to quote some of its values:

Mg

406K

Fe Cu

467K 399 K

Cr

Ag

402K

225K

At very low temperatures, and also in some other cases, metals exhibit depar-

tures from the 7° law. Explanations will be furnished in Chap. 3. In spite of the excellent agreement with experiment, Debye’s theory cannot be regarded as rigorous, because of the simplifications on which it is based. One

of the major simplifications is the choice of a quadratic dependence for the spectral density D(w), which may differ radically from its true form (except in the

range of very small frequencies). As an illustration, Fig. 2.14 presents D(w)

2.2

Dw

The Specific-Heat Capacity of the Lattice

117

D(wmax)

04

08

12

16

>

2W/ingx

Fig. 2.14. Spectral density D(w)/D(w,,,,) a3 a function of w/w,,,, for a copper crystal, calculated by the root-fitting method for a model taking account of the interaction between atoms up to the second coordination zone (Ref. 2.1, Fig. 9]: (/) branch of longitudinal oscillations, (2, 3) two branches of transverse oscillations; F stands for features arising from nonsingular critical points, and S for those due to the usual saddle points

functions calculated by Owerton [2.5] using the root-fitting method. In the figure

curves are shown for three acoustic branches of copper, the letters F and S

marking the singular points of the function D(w). From the curves depicted in Fig. 2.14, it is seen that their initial segment is well approximated by the Debye parabola, showing that Debye’s theory is in agreement with experiment for small frequencies. Any attempt to compute the function D(w) reduces to a problem of numerical computation. This compels one to proceed from the general equations (2.1.37). However, qualitative assumptions as to the form of the function D(w) can also be made on the basis of purely topological theorems, without numerical computation. Specifically, the existence of critical points in the family of surfaces @, = const in qg space and of a singular function D(w) can be shown to be a necessary consequence of lattice periodicity. This problem was first explored by van Hove [2.6] (van Hove’s theorem). We do not intend to enlarge upon this problem, but we illustrate the method

for the case of one spectral branch, following Ziman’s treatment [1.30]. According to the definition of the function D(w) (2.1.14, 2.2.13), we may set D(w) do = (v,/8n3) § dq .

(2.2.18)

The integration in (2.2.18) is performed over the volume of the layer in g space, where the frequency «, is contained in a narrow interval w < w, < w + dw. The integral in this equation may be transformed if the integration element is chosen not in the form of dq, dq, dq, but as an infinitely small cylinder whose lateral surface is perpendicular to the surface w, = w, with the area of the base being dS,, and the height dq, = dw/|grad w,| = dw/|v,|

.

(2.2.19)

118

2. Dynamic Properties of the Crystal Lattice

Here grad w = {(dw,/0q,), (dw,/ 0q,), (Om, /0q,)} is a frequency gradient, that is,

a vector which has the dimensions of velocity (in the Debye model this is the

constant »,), the direction of the vector coinciding with the normal to the surface @, = w. As a result of this transformation, (2.2.18) becomes

D(w) dw = (v,/8n3) ff dS,, dq, .

(2.2.20)

Substituting (2.2.19) into (2.2.20), we find

(2.2.21)

D(o) = (v./82°)§ (dS../\U4l) -

Formulas such as (2.2.21) will ascertain the singularities of the function D(w), and later on they will be employed to calculate the density of electronic states in metals (Chap. 4). Singularities evidently should arise at some critical points ¢,, for which the

quantity », goes to zero; i.e., the frequency w, at these points turns out to be a locally “planar” function. Consider this function in the vicinity of a critical point, expanding it there in a power series of the difference y = g —,,. There is no linear term in this expansion since, by the definition of the critical point, at such a point o, = V,w, = 0. The quadratic terms, after being brought into the normal form (upon transformation to the principal axes), will reduce to a sum of squares

Wg = We t

Yi ta,ye+asy3t+...,

(2.2.22)

with a, = 40? w,/dy? (i = 1, 2, 3). If all @, < 0, the function @, at point g,, has a maximum

and the surfaces w, = const, according to (2.2.22), have the form of a

family of ellipsoids. The volume of an ellipsoid of this family for the frequency @, = w with the surface surrounding the point q,, is equal to

(41/3) (w., — oP? /|o, 203 |"? . The differential of this volume when density

multiplied by v,/8x° gives the spectral

D(w) = (v,/42?) (Wp — )"/? [any argay |"? ,

(2.2.23)

where w < w,,. It is clear from (2.2.23) that this singularity does not perturb the continuity of the function D(q) itself, but its derivative 0D(w)/dw has a discontinuity, tending to — co when w — w,, from below. A similar situation occurs

for the minimum @,. Figure 2.14 portrays the different singularities of D(w) for the acoustic branch of the spectrum of copper. If one of the a; has a sign opposite to that of the two others, a saddle point appears for which the singularity of the derivative @D(w) / dw arises again. We

demonstrate this for a frequency spectrum by reference to a planar lattice model.

2.2

The Specific-Heat Capacity of the Lattice

119

42 A o!

A, 0”

A oo

38 A of

v8 A

q i ke a

5

a

C

bs

ly

A > °

5

ds

05

Ag

Fig. 2.15. Searching for the saddle point in wave-number space

Shown in Fig. 2.15 are several unit cells, that is, zones in g space (q,, q2). The function w, is considered to be periodic and continuous. It may therefore be

expected that each cell will have at least one maximum

(clear circlet A,, A>,

A;,...) and one minimum (dark circlet B,, B,, B;,...). If we connect the maxima of closely adjacent cells, for example A, and A,, by a curve (/, in Fig. 2.15), the latter will have at least one point at which the function w, takes the smallest value for this curve. Similar points will be present on any other

curve that connects the maxima A, and A, (for example, on curves //,, II1,).

The locus of all these points generates a continuous curve B,B,(J,) that

connects two adjacent minima of the surface w,. On this curve there will be a

point C at which the w, assumes the largest value on this curve. This point should be a saddle point, since on the curve /,, when we move from B, to Bs, it will correspond to a relative maximum, and, if we move along curve I], from A, to Ag, it will correspond to a relative minimum. The same applies to the curves which connect points A, and A,, B, and B,. Therefore, the function w, has at least two saddle points in each cell. This line of argument certainly becomes more complicated if we take into account all the branches of the spectrum, but the gross features of the treatment remain the same. For small q the dispersion relation is linear: w ~ q i.e., dw, /0q = v; = const.

Then it follows from (2.2.21) that when w

uy

Localization of Phonons on Point Defects

d

*d

125

(2.4.2)

= 5, J dqU(a)exp(—igl) . —rid

Multiply (2.4.1) termwise by exp(iq/) and calculate the sum over /. This gives the following: .

dA

m da + A6d;,o)ipexp(iql) = — mo| Ua) +

*é

J aquca)|

(2.4.3)

—xid

— aD, (Quy — Uy44— 4 4)exp(iql)

= —a) [2 —exp(igd) — exp(—igd)]u,exp(iq!) = —mw2U(q). t

— (2.4.4)

Equating (2.4.3) to (2.4.4), we find

wg -oru

= (32dA\ ) "I§ dqU(q). —xid

(2.4.5)

Consider two limiting cases. First, let aid

J

—xid

dqU(q)=0.

(2.4.6)

Then U(q) #0 only if w? = w? = (4a/m)sin?(qd/2). For w? > 4a/m such solutions do not exist at all, and for w? < 4a/m there are two solutions: q = + qo,

Go = (2/d)arcsin[(mw?/4a)!/?]. Condition (2.4.6) yields

U(q) = A[6(q + 40) — 5(4— 40)] ,

(2.4.7)

where A is an arbitrary constant, and 6(q) a Dirac 6 function. As is clear from (2.1.5), the presence of a defect under (2.4.6) does not in any

way

change the spectrum of these oscillations in comparison

lattice. Let now, in contrast to (2.4.6),

(d/2m) ‘{ dgU(q)=C #0. Rid

with an ideal

(2.48)

Then (2.4.5) immediately yields

U(q) = w? AC/(w? — w?) ,

(w*dd/2n) “ (w2—w2)-'dq = 1. —x/d

(2.4.9)

(2.4.10)

126

2. Dynamic Properties of the Crystal Lattice

Introducing the notation w? = 4a2?/m, qd = x, rewrite (2.4.10) in the form

(224/2n) f (sin?(x/2) —Q?)"'dx =1.

with

(2.4.11)

In solving (2.4.11) for Q? < 1, we encounter a difficulty which is associated

the nonintegrable divergence of the integrand

for sin(x/2)= +2.

To

eliminate the divergence, we add to the right-hand side of (2.4.1) the frictional force (proportional to the velocity)

ii =—ymu, = iwymu, .

This

leads

to

the

(2.4.12)

replacement

w*>w?+iwy

or

27-+27+iQn

(q = y(4c/m)~'/?). Then we should let 7 approach zero and make use of the identity [2.11]:

J (p(x)— in) * dx|, 5 =S'dx/(x)+ inf dxd[o(x)] ,

(2.4.13)

wheref denotes the integral in terms of the principal value. Calculation yields = (1/2n) F (sin?(x/2)— Q?)~ 1 dx = (2nQ(1 — 97))~ 1/2

tg(x/2)— Q(1— 0?) 7

|[4*

/02)) ||_ =O O0 +x ig( Q?)'? . (i/2) i dx6(sin?(x/2) — Q?) = iQ(1 —

(2.4.14) 2415)

We assume here that Q? < 1. Substituting (2.4.15) into (2.4.11), we find iNA/(1—Q?)'"2=1,

Q2=(1—A?)',

(2.4.16)

which contradicts the initial assumption Q? < 1. Then we try a solution of (2.4.11) for 2? > 1. The integral in (2.4.11) is easy to calculate, and we obtain

(1/2n) , [sin?(x/2) — 22] -! dx = — 1/Q(Q? — 1)! , AQ +(Q?2—1)'2 =0. When

4 > 0 this equation is incapable of solution, and for 4 < 0 (ie., fora

lighter isotope)

Q=(1— 42)? ,

(2.4.17)

2.5

Heat Capacity of Glasses at Low Temperatures

127

Thus, in addition to the solutions with 2? < 1 which existed in an ideal crystal, a

split-off level has arisen which possesses the frequency wW = Wma,(1 — 47)~ 1/2 > Wmax(Wmax = 2(a/m)"?). It follows from formula (2.4.9) that U(q) at o=a@o does not display singularities (as a function of q) anywhere on the real axis. As is known from the Fourier theory [2.12], the integral (2.4.2) for u, decreases quickly to zero when | > oo. Thus, (2.4.17) yields an expression for the frequency of a localized phonon, i.e., a lattice vibration mode in which the displacements u, are localized near a defect. The possibility of the occurrence of such vibrational modes is one of the most important features peculiar to the spectrum of a real

crystal as compared to that of an ideal one.

2.5 Heat Capacity of Glasses at Low Temperatures As already noted in Sect. 1.10, modern solid-state physics is by no means concerried with crystals only. Of great interest are the properties of amorphous

solids (glasses). Naturally, the question arises: Which of the above results may be

applied to glasses and, primarily, are there phonons in them? This problem is

nontrivial. As has been emphasized in Sect. 1.10, it is altogether pointless to

describe the states of glasses in the language of wave functions—some averaged characteristics are needed—and it is therefore not even clear how the problem of elementary excitations in glasses and other disordered systems should be posed at all. An adequate mathematical device is the resolvent (Green’s function) method, which will be considered in Chap. 4.

All these difficulties, however, arise in attempting to consider excitations

similar to localized phonons (Sect. 2.4) or short-wavelength phonons (which in a

disordered system, generally speaking, may even not be specified by a definite quasimomentum gq). Everything is much simpler for long-wavelength phonons.

The point is that when considering them we may approximate the real atomic structure of a glass as an isotropic elastic continuum [2.13]. In the latter, as can be shown purely phenomenologically, longitudinal and transverse sound waves exist with definite frequency and wave vector, w and qg, with w ~ q. In keeping with the general principles of quantum mechanics, one can assign quasiparticles

to these waves with a momentum hg and energy hw. As with crystals, it is quite

natural to call such quasiparticles phonons. They also obey the Bose-Einstein

statistics. An immediate consequence might seem to be that at low temperatures,

when excitations with small w and q are substantial, crystals and glasses should not differ in thermodynamic properties and the Debye T> law for the heat capacity C, should hold for glasses. However, experiment shows that at very low temperatures practically all glasses exhibit a linear T dependence of Cy. As will be shown in Chap. 3, the same linear temperature dependence in metals

is due

to conduction

electrons.

This

linear term

in the lattice heat

128

2. Dynamic Properties of the Crystal Lattice

capacity can be separated out directly in dielectric and semiconducting glasses, but in metallic glasses it can only be separated out at temperatures below the superconducting transition temperature if this exists. As follows from the above arguments, this dependence does not have to be due to phonons. This implies

that there may exist in glasses some extra specific excitations that are absent in

crystals. A model for such excitations, called two-level centers (or tunnel states), was proposed by Anderson et al. [2.14], and also Phillips [2.15].

As already mentioned in Sect. 1.10, in contrast to a crystal, a glass has no uniquely defined structure. Anderson et al. and Phillips (2.14, 15] suggest that in glasses there exist atoms (or groups of atoms) which can occupy two positions

almost equivalent in energy. This double-well character of interatomic poten-

tials, which arises quite legitimately, say, in hydrogen-bonded crystals (Sect. 1.7), may “fortuitously” occur for some atoms, whatever the type of bond. For a detailed discussion of this assumption see [2.16]. Let it be noted that the shape of the potential with two minima (as a function of one of the x coordinates) differs drastically from a parabolic one. Thus, we deal with regions in which the anharmonic effects are anomalously large; i.e., the structure is highly softened. Three-well and more complicated potentials are apparently less probable than two-well potentials and, therefore, do not seem to make any substantial contri-

bution to the thermodynamic properties of glasses. The problem of the energy spectrum of a quantum system with a potential having two minima was solved in Sect. 1.7. If E, and E, are the energies of the

lowest levels in the first and the second well, the distance between the two lowest energy levels of a two-well system is equal to (Sect. 1.7)

(2.5.1)

AE = [(E, — E,)? +|V,2|7]' ,

where V,, is the matrix element of the Hamiltonian describing the tunneling of an atom from one well into the other. Since the atomic mass is very large, the

tunneling probability and, hence, the quantity V,, are extremely small. The energies E, and E, may take different values, but we are concerned with the case E, = E, and therefore the quantity 4E is small, too. Let us consider the problem of the contribution which a two-level system with an excitation energy 4E ~ kgT makes to the thermodynamic quantities

(higher excited states possess energies that are much larger than kg7 and their

contribution is exponentially small). According to the Gibbs distribution, the

mean energy of a two-level system with energy levels ¢; (i = 1, 2) is

é=

ye

exp(—¢;/kgT)

d exp(—¢;/kpT)

AE

~ exp(4E/kyT) +1’

(2.5.2)

where the lowest level is taken to be the zero of energy ¢, = 0; then e, = 4E. The

2.5

Heat Capacity of Glasses at Low Temperatures

129

relevant contribution to the heat capacity is then equal to

(4E?> — exp(4E/kgT)

CANS MES Bg (exp AE ha) + WF

(2.5.3)

Thus from (2.5.3), the contribution to the heat capacity from a two-level defect

(Schottky contribution) goes exponentially to zero for kgT< AE, falls off as T ~? for kgT > AE, and has a maximum at intermediate temperatures. But in glasses, according to the model discussed, there exist two-level centers with diverse excitation energies, including very small onés. To get a final answer, we

need, in addition, to average formula (2.5.3) over all values of 4E with the corresponding density of states D(AE).:

(2.5.4)

Cy = f d(4E)D(4E)C,(AE) . 0

It is assumed that D(0) # 0; i.e., there is a fairly large number of two-level centers

with excitation energies as small as desired (to be more exact, with AE < k,T). Of course, as follows from (2.5.1), the quantity 4E may not be less than the minimum value |V,,|; however, as noted above, the latter is apparently very small. If kg Tis small compared to the representative splitting value (4E)

assigning

the scale of decay of the function D(AE) with increasing AE, the D(AE) in (2.5.4) may be replaced by D(0). Then, upon replacement of the variables 4E=kgTx we have

~ k2TD(0) fdxxtet Sean op

2

7 D(KET

(2.5.5)

Integration by parts allows the integral involved in (2.5.5) to be reduced to the form

1=2 5 (19! f de

one

and using the replacement of the variables nx = y, we have

= 25 d

ry yo cr"

.

130

2. Dynamic Properties of the Crystal Lattice

Since wo

J dye

_

wo y

=>

yal,

( _

4

1y" +1

n?

n? =

12’

we obtain the result that

2 dxx?e* =x? j =—. o(e*+1)? 6

(2.5.6)

Thus we have obtained the required linear temperature dependence of heat capacity for kg T to, one does not have to allow for the contribution of these centers to D(0), since during the experiment the atom may be regarded as localized in one of the wells; it simply does not have the time to become “smeared” over the two wells. As is known from quantum mechanics, the quantity t,, depends exponentially on the spacing d between the wells and on the height U of the barrier: T 2 ~ exp(./2MUd/h)

,

(2.5.7)

with M being the mass of a tunneling atom. Here U and d are distributed approximately with a constant density, and the number of centers for which T12 d (with c being the velocity of light), so that the photon wave vector may to a high degree of precision be set equal to zero; i.e., the electric field E may be assumed to be homogeneous. Let us consider a linear two-ion array with ions of mass M and charge +e, and with ions of mass m and charge — e. The equations of motion will then take

the form (2.1.18)

MU = —2a(U—u) + eEexp(—iwt) , ma = —2a(u—U)—eEexp(—iwt)

(26.1)

,

where, in accord with what has been stated above, g = 0; this corresponds to Wmex (2.1.27). According to the well-known electrodynamic relation,

E=D-4nP, where D

(2.6.2)

is the electric induction,

P = ne(U—u)

(2.6.3)

is the polarization vector of the medium, and n

is the number of unit cells in unit

volume. The first term in (2.6.2) describes the field produced by external forces, and

the second the long-range (Coulomb) contribution of the ion interaction which

is taken into account in the mean-field approximation (2.6.2). On the other hand,

the short-range contribution of the interionic interaction is described by the term that contains « in (2.6.1). Trying a solution of (2.6.1) in the form

U(t) = Ue

,

u(t) = wei"

and combining (2.6.1), we find

MU+mu=0.

(2.6.4)

Substituting (2.6.4) into (2.6.1) yields

U = ecE/M(wi-w’)

,

(2.6.5)

132

2. Dynamic Properties of the Crystal Lattice

where @p is defined in (2.1.27). The polarization P, according to (2.6.3, 5), is equal to

M “j= P = neu(1+™)

fit E —+4+—nora ne( apt

.

( 2.6.6 )

The permittivity ¢(w) is determined (2.6.2) by the formulas

D=c(w)E

or

P=([e(w)—1]E/4n .

(2.6.7)

Comparing (2.6.6, 7), we find

(2.6.8)

£(@) = 1+ @} /(w5 — 0”) . The notation introduced here is

w2 = tnne?(

+ )

.

(2.6.9)

The e(w) relation is schematically represented by a solid line in Fig. 2.16a. The expression for e(w) is inapplicable in the vicinity of wo, where e(w) — oo.

Ew)

a

é) | | '

!

| 0

®

@

Fig. 2.16. Schematic frequency w dependence of (a) the real c’ and (b) the imaginary e” part of permittivity

2.6

Allowance

High-Frequency Permittivity of Ionic Crystals

133

for the frictional force (2.4.12) leads to the replacement w? > w?

+ iyw (Sect. 2.4). The result for the real and imaginary parts e(w) = e'(w) + ie” (w)

(Fig. 2.16) then will be

e'(w) = 1 + w2 (we — w?)/[(w3 — w”)? + y?w?] ,

(2.6.10)

£"(w) = yous /[(w§ — w?)? + y?w?] . The

maximum

of absorption

sponds to the optical-phonon “optical”).

[determined

frequency

by e”(w)] [2.17] at small y corre-

w,

(which justifies

the attribute

As is known from the electrodynamics of continuous media [2.17], longi-

tudinal (L) and transverse (T) electromagnetic waves may propagate in a system.

The frequencies of these waves are determined by the equations (for y = 0)

&(,)=0,

ow

=(@2 +)?

wze(wz)/c? = q? ,

,

(2.6.11)

c?q? = wi[1+032/(w§—@})] .

(2.6.12)

Since the thermal motion of ions is neglected, the frequency of longitudinal waves turns out to be q independent. If we turn off the interaction of the lattice with the electromagnetic field (e +0), (2.6.12) yields solutions (Fig. 2.17a)

O=W,

Wr=cq

(2.6.13)

describing a photon and an optical phonon that do not interact. An interaction leads to mixing (hybridization) of the modes (Fig. 2.17b). For @ , and the free neutron have a wave vector k and, accordingly, a wave function

W,(r) = (22) > exp(ik-r)

(2.7.1)

(normalized to a 6 function). As a result of the interaction of the neutron with the crystal, the latter passes to a state |¥,>, and the former to a state |Wy>. The

probability of this process Py _,,4: may be calculated from the time-dependent quantum-mechanical perturbation theory (Sect. 1.5), on the assumption that the potential of the interaction of the neutron with the crystal, V, is small compared

to the characteristic excitation energies [1.12],

1 o i d Py on = iz J arexe| jE, — E, + & — & )t] oo

XCF

We VIP de> Pde 5

(2.722)

2.7

Lattice Scattering and the MGssbauer Effect

135

where E and « are the energies of the corresponding states of the crystal and neutron, and dv,, is the density-of-final-states differential of the neutron. Represent V as

v=) o(r—-r,) ,

(2.7.3)

7

where

(2.7.4)

r= Ritu,

are the coordinates of the jth ion, R, is its equilibrium position, and w, is the

displacement vector. For simplicity assume that all ions in the lattice are alike

and the form of the function v(r — r,) does not depend on j. Substituting (2.7.1, 3) into the matrix element V, we find

CPV l VP ibad =

d

en “ Y,exp(—ig-r,) are introduced. The operator ny is the Fourier component of the ionic-density operator in the crystal n(r) =P d(r—r,) , I

Next, we make

ng = ((2x)~*)

f drn(r)exp(—ig-r) .

(2.7.6)

use of the identity

exp| iE, - en |¢ P,|ng|P,> = CP lexp(it/h)n, exp(—ist/h)| P,> In (Ol. = CF,

,

(2.7.7)

where 3 is the Hamiltonian of the crystal and n,(t) is the Heisenberg rep-

resentation of the operator n, [1.12]. Here we have taken into account that # |\¥,.> = E,|%,,.>- Substitution of (2.7.5, 7) into (2.7.2) yields

136

2. Dynamic Properties of the Crystal Lattice

AP unk = I73 J drexplioo)|C¥,ing(0)1%1> logl? avy 1

wo

©

.

= pe J dt J drexplio(t—0’))< ¥iln_,(O1P.)

(2.7.8)

x CP ylng(t)|P1) lvgl2dvy. where we have introduced the notation w=h"'(e,.—«,) and extended the square of the modulus of the matrix element allowing for the readily verifiable relation

CP alnglPid* = CPi ln_ gl Pad A direct observable is the transition probability per unit time, summed over

all final states of the crystal (with ),|¥,> (h/2me,)'/2(6, — b* Jexp(iql) . q

(2.7.19)

For a three-dimensional crystal, just as for a linear array, the crystal oscillations in the harmonic approximation may be represented in terms of an assembly of independent oscillators specified by the numbers v = q, s, where g is the wave vector, s= 1, 2,..., 3a is the branch number, and o is the number of atoms (ions) per unit cell (2. 1. 3). Here we introduce the operators bt and b,, which satisfy commutation relations such as (2.1.47)

(6,62). =6,,,

[6,6]. =(67,62]_ =0,

(2.7.20)

and are related to the Hamiltonian # and the displacement vector #; in the same way as they are in (2.1.46, 48):

H = J ho, (bf 6, +4) = Ey + hw,be b,

(2.7.21)

2.7

Lattice Scattering and the Mossbauer Effect

4, =i D (h/2Nmo,)"!7e, (bys — 5%, Nexplig: Ry) .

139

(2.7.22)

where e, is the corresponding matrix eigenvector G (2.1.35) normalized to unity \e,] = 1 (called the polarization vector). Since we assume that the lattice is made up of like atoms (ions), m, =m is independent of s. It is no trouble to derive

formulas (2.7.20—22), which are written by analogy with a linear array (we leave it as an exercise for the reader to prove this). More

important

is the mani-

pulation of such Hamiltonians and operators since they arise in a wide variety of solid-state theory problems that might seem to have no bearing on lattice dynamics (Chap. 5). To start with, we calculate

By (t) = exp(iof t/h)bs exp(—is t/h) .

(2.7.23)

Since, according to (2.7.20), [67 , bt b,.]- =0, [b¢ b,, bt 6,.]_=0 when v 4 v, we obtain

bt (v= exo(

y

vty

hw, bt 6, expt:

x bt exp(—iw,tb+ bern (—F

= ean ZS,

b,) Y

vty

ho,.bt é.)

hw, bt 6, Jexp( - id. hw,.b* 6,)

(2.7.24)

x exp(iw,tb? b,)bt exp(—iw,tbt b,) = exp(io,tb* b,)bt exp(—iw,tbt b,) . Thus, we need to calculate

B+ (a) = exp(abt b,)b* exp(— abt b,) .

(2.7.25)

Differentiating B (a) with respect to «, we find

dB? (a)/da= exp(ab* b,)b* b,b+ exp(— ab? b,)

= exp(ab, b,)[5y 6,, 6, ]_exp(— ab; b,) = exp(aby 6,){by [b,, 67 1- + [7 , 6. 1-6,}

(2.7.26)

x exp(— ab, b,) = BF (a) ,

where we have made use of the identity

(AB, C]_ = A[B,C]_+[A,C]_B,

(2.7.27)

140

2. Dynamic Properties of the Crystal Lattice

which is proved by direct expansion of the right-hand and left-hand sides, and employs (2.7.20). Equation (2.7.26) may be integrated with the initial condition By (0) = b+. Then, by virtue of (2.7.25),

By (a) = bt expa .

(2.7.28)

The possibility of extending the usual methods of solving equations for functions to operators is, generally speaking, nontrivial. In the present case it may be achieved as follows:

Bs (a)= Bi (0) +.

2 B+

da

ja=0

+424 By da?

ja=0

= BY O)(1+a+4a2+...)=brexpa, where the function of the operator is expanded in a Taylor series. Comparing (2.7.23, 25) and allowing for (2.7.28), we find

bt (t) = explia,)b? .

(2.7.29)

Similarly, we may find

b,(t) = exp(—iw,t)b, .

(2.7.30)

Substitution of (2.7.29, 30) into (2.7.31) yields

j(t) = i Y(h/2Nmo,)"? e,exp(ig: Rj) x [b,,,exp(—iw,t)—b*,, exp(iw,t)] .

(2.7.31)

Note also that for any operator A we have

A"(t) = exp(iof t/h) A"exp( —i Jt/h) = exp(ix t/h) Aexp(—is?t/h)exp (io t/h) A

x exp(—isf t/h)exp (ist t/h) ... Aexp(—i# t/h) = [A(T and hence for any function f(A) prescribed by a Taylor series

exp(it t/h) f(A)exp( — if t/h) = flexp (iF t/h) Aexp(—ivf t/h)) .

(2.7.32)

Now we are aware of what n,(t) is. Let us turn to the problem of calculating the mean values of the products of the operator bt, 6 according to (2.7.14),

2.7

Lattice Scattering and the Méssbauer Effect

141

where @ is defined according to (2.7.13), and 9 according to (2.7.21). It follows from (2.7.13, 25, 28) that

07 bY @ = Bt (Bho,) = by exp(Bha,) , Le.,

(2.7.33)

bso =exp(Bha,)ob; . Similarly,

(2.7.34)

be = exp(— Bho,)Qb, . Then

Tr([b,, bf ]- @) = Tr(b,by @) — Tr(by be) = Tr(b, 96; Jexp(Bhw,) — Tr(b? b,e) = Tr(by 6, o)exp(Bhw,) — Tr(b; be) = Tr(by be)(exp(Bho,) — 1) ,

(2.7.35)

where, just as in the derivation of (2.7.12), we have used the freedom to perform a

cyclic permutation of the operators under the trace sign. But [b, ,b,]_ = —1, Trg =1, and Tr(by b,e@) = , and therefore we find from (2.7.35)

(2.7.36)

= (exp(Bhw,) —1)-!

Thus, we have derived the formula (2.1.49) for the mean of the occupation number in the Bose-Einstein statistics. A similar technique will be used for calculating the average of an arbitrary number of phonon operators A,,

Az,..-. A, (each operator A; is either b7 or b, [2.19]). Then, according to (2.7.20), [A;, A;]- is a c number (i.e., commutes with any one of the operators

b+ , b,). Therefore,

[A,, 4,43... Ag] = Ay Az... A, — AQ Ag... Ag Ay =[A,,4,]-A3...A,+A,AyA3...4,— AQ Ay... Ay Ay =[A,,A,]_A3.-.4n + (41, 43]- 42 4g --- An + A,A3A,...4,—A,A3...4,4y = .-= YUAL

Aid

A.

Ava

Aves

110

Ay.

(2.7.37)

142

2. Dynamic Properties of the Crystal Lattice

Now we average the right-hand and left-hand sides of (2.7.37):

Tr([A,, Az.

A,]- 0) =Tr(A, A, ...A,@) — Tr(A, Az ....A, A; 0) = Tr(A, Az .. .A,Q)

—exp(+ Bhw,)Tr(A,

= (1—exp(+ Bhw))

(A, A3..-Ag-1) -

(2.7.40)

The averages involved in (2.7.40) may be transformed further proceeding in the same way:

(A, A, A3 Ag) = (A, Az) (A3 Ag) + (A, A3) 4A? Aa? + (A, Ag) ¢A2A35)

ete.

We

,

have arrived at the result that an average

such

as that involved

in

(2.7.40) (with n being even) is equal to the sum of all the products of the averages of the pairs of operators involved in the product A,, A2,..., A,, the operators in each pair being taken in the same order as in the initial product. For an odd n it may be shown in an analogous fashion that = 0 (since ¢A,>

= 0). The statements formulated above also hold true in the case when the A, are

arbitrary linear combinations of the operators b* and b,. The result obtained (normally referred to as Wick’s theorem) was proved by Wick for the averaging

over the ground state [2.20], and by Bloch and de Dominicis for a canonical ensemble [2.21]. Two important relationships were used here: (1) the com-

mutator of any two operators b*, b is a c number, (2) the commutator of any

operator b*, b with the Hamiltonian is the same operator (to within the factor + ha»), which results in identities (2.7.33, 34). When the anharmonic terms in the

2.7

Lattice Scattering and the Mossbauer Effect

143

Hamiltonian are taken into account, this is invalid and Wick’s theorem does not

hold any longer.

Now we are in a position to calculate the dynamic form factor, having learned in passing some operator handling procedures which will be employed

in Chap. 5.

2.7.3 Calculating the Dynamic Form Factor in the Harmonic Approximation We assume that the crystal is described by a harmonic Hamiltonian (2.7.21). We

need to calculate the quantity

S(q, w) = (2x)® f dtexp(iwt)

x ¥ expLig (Ry —R,))

(2.7.41)

[we have substituted (2.7.4, 6) into (2.7.15)]. The expression to be averaged in

(2.7.41) can be readily represented as the exponent of some operator (if [A, B]_ #0, then e4e® #e4**), Find the commutator of the operators —ig@,(t) and iga,.. Inserting (2.7.31) and using (2.7.20), we obtain : oa 1 [- ig: &,(t), ig: a;-) -= N y (h/(2m(w45@y5)"!?)

kK ss

x exp[i(kR, +k’: Rj) 19° ees)(Q° ex's) [bas ExP( — ig; ¢)

—b* 45 expliogs t), Bey — bt ey ]=

i

Y

N kk ss”

A(2m)~* (ops Op)?

x exp[i(k-R, +k’

Ry-))(qeas )(Q exes)

(2.7.42)

X [5p,—4° Sas: CXP( — igst) — Se, — 4 Sos EXP(IO5 t)] 1

.

=N X(h/2may, exp(ik (Rj — Ry ))1q- ees |? ks

x [exp( — ico, t) — exp(ia,, t)] (we have employed the relation e_,,, = ef, which may be obtained from an exploration of the properties of the matrix Y). The commutator of the operators with which we are concerned is a c number. In this case,

exp Aexp B = exp(A + B)exp}[A, B]_

.

(2.7.43)

144

2. Dynamic Properties of the Crystal Lattice

This equation may be proved as follows. Introduce the operator S(A) = exp(—AA)exp/(A + B) .

(2.7.44)

Then dS(A)/dA = — exp(—AA)Aexpd(A + B) + exp(—1A)(A + B)exp A(A + B) = exp(—AA)Bexpd(A + B) = exp(—AA)Bexp(AA)S(A)

(2.7.45) .

In its turn, < exp(— 2A) Bexp(AA) = —[A, B]-

,

exp(—AA)Bexp(AA) = B—A[A, B]_

.

ie. (2.7.46)

Substitution of (2.7.46) into (2.7.46) yields

dS(A)/dd = {B — ALA, B]_}S(A)

.

(2.7.47)

The solution of (2.7.47) becomes 1

S(1) = exp| fain — ALA, B]_ | = expBexp(4[A, B]_) . 0 The possibility of applying the conventional method

(2.7.48)

of solving differential

equations to (2.7.47) is due to the [A, B]_ being a c number and may be proved in about the same way as (2.7.28). Thus, it remains for us to find

= ceno( E02 Nmonys)''? (4: ee,s) x { rexpatk -R,.) — exp(ik: R;— inst) ] Pas —[exp(ik: R,.) — exp(ik- Rj + ian t18°,4))

= expC) .

(2.7.49)

Since the operator involved in the exponent is a linear combination of the

2.7

Lattice Scattering and the Méssbauer Effect

145

operators b, b*, Wick’s theorem is applicable to it: +a

. However, the situation is not so simple as that: when the first nucleus emits a gamma quantum, this photon

carries

away

not

only

the energy

E,

but

also

the

momentum

E,/c,

and

accordingly, the first nucleus recoils and receives some kinetic energy. That is to say, the energy carried away by the gamma quantum is not E, but E < Ey. The latter may be determined from the laws of conservation of momentum and energy

Eo=E+p?/2M,

Ejc=p,

(2.7.76)

{_—_—

0—

t

Fig. 2.19. Resonance absorption without nuclear recoil.

with M being the nuclear mass. Using the condition Ey < Mc? (say for the classical “Méssbauer” nucleus *"Fe Ey ~ 1.4 x 10*eV, Mc? =6 x 10'° eV), (2.7.76) yields

E=E,-R,

RE/2Mc?.

Similarly, for the second nucleus to absorb the gamma

(2.7.77) quantum, it must

possess the energy E, + R. The absorption (emission) probability, as is known from quantum mechanics, has the form depicted in Fig. 2.20a, where I is the

natural width of the level |1 > (’ = h|t, with t being the lifetime of the nucleus in the state |1), i.e. the inverse probability of the |1)>—|0> transition per unit time). Then, as is seen from Fig. 2.20b, when I . The probability for

the nucleus j to have the value of the momentum hk here is equal to

we = ICKL[2 = [f drjexp(—ik-r,)/(2n)?? x Pry...

stp.

tw I? ,

(2.7.78)

where ¢k|l> stands for the expansion coefficients of the wave function |/) in plane waves |k> with r =r, (2.6.16):

>= Zed should be

If> = lk—-9@> =exp(—ig-r,)|k> and

the completeness condition yy |k> must be

expanded in the eigenstates of the nucleus |n>, and the probability of the transition during the emission of a gamma quantum is

(2.7.82)

-

DP? Prag =n f>P =lCnlexp(—ig-e

+n

If, as a function of its energy E, the intensity of the gamma quantum emitted is equal to [,(E) (Fig. 2.20a), then the intensity of the gamma quantum emitted

during the /-+n transition will be [9(E + E,— E,) (so that, in keeping with the law of conservation of energy, the maximum shifts by 4E = E, — E,). The total

intensity, upon averaging with respect to the initial states and upon summation over the final ones, is

I(E) = J) Io(E + Ey — E,)| 7 dt(2nh)~ ' exp(—iEt/h)Io(t)@, al

-a@

x Cllexp(iE,t/h)exp(—ig-r,;)exp(—iE,t/h)|n>

x d), the distances between adjacent levels Deviations will The problem To each value modulus of the

are very small and the spectrum is practically continuous. be observed for very small particles [3.5], when L2d. under consideration also possesses a high degree of degeneracy. of the energy ¢ which, according to (3.5.4), depends on the vector p, there corresponds a set of wave functions (3.5.3) that

depend also on the direction of the vector p. Therefore at the outset we must

determine the degree of degeneracy g(«). To this end, we calculate the number of

states in which the electrons possess the value of the momentum

within the

limits from p to p+ dp for all possible directions of the vector p. To these states in

P space there corresponds a set of phase points that fill up a spherical layer of volume 4np*dp (Fig. 3.5), which, from (3.5.4), is equal to

DS l2g_m*3/2p1/2do

(3.5.5)

dp

Fig. 3.5. Calculation of the degree of degeneracy of the Fermi gas, g(«)

172

3. Simple Metals: The Free Electron-Gas Model

To find g(e) we have to divide by this unit phase cell volume 4p, Ap,

is determinable from the Heisenberg uncertainty principle

AxAyAzAp, Ap, 4p,~h? .

Ap,, which

(3.5.6)

The quantity 4xAy4Az involved in (3.5.6) gives the accuracy of localization of an

electron in normal space. In the present case it is equal to the volume of the metal,

V, since what counts here is that the electron is in the metal. Then,

according to (3.5.6), the momentum determination accuracy, i.e., the size of the unit phase cell in p space, is

Ap, 4p, 4p, © h?/V

.

(3.5.7)

Such a derivation of the unit volume in phase space does not give the correct numerical coefficients. However, a straightforward calculation of the number of states such as that performed in Chap. 2 will lead precisely to (3.5.8). To allow

for the spin degeneracy (in the absence of a magnetic field and spin forces), the

quotient of the left-hand side of (3.5.6) by the right-hand side of (3.5.7) is multiplied by 2. This yields g(e) = 5 4n(2m*)?/? Vell? = Cell? ,

(3.5.8)

To find the distribution function f(¢) of electrons in states with different ¢, the degree of degeneracy (density of states) should be multiplied by a statistical

quantity, ie., the mean with respect to the occupation number of states with

different ¢—n(e). The mean occupation

numbers

for noninteracting

particles obeying

the

Pauli exclusion principle are determined by the Fermi—Dirac distribution function. The latter may be derived in a fashion analogous to the treatment of twolevel systems in Sect. 2.5. Let {v} be a one-particle quantum state, and ¢, the corresponding energy. The total energy of any quantum state of a system of many noninteracting particles then may be determined by specifying occupied and free states

&=Sen,,

(3.5.9)

where n, = 0 for free states and n, = 1 for occupied ones. In contrast to Sect. 2.5, we explore here a system with a prescribed number of particles

N=Yn,.

(3.5.10)

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

For

this reason we should

use a large canonical ensemble,

173

introducing a

chemical potential ¢. Since we consider the electrons to be noninteracting and

B c d p a g s } “zalh eT

each state v to be filled up independently, we have

that is,

nce) =| exp( SF) + i] B

(3.5.11)

Thus the function f(e) for the quadratic dispersion relation (3.5.4) has the form

Se) = gle)n(e) = 4n(2m*)>? Vh-3/fexp[(e—0)/kyT] + 1}

(3.5.12)

The chemical potential ((7) is determined from the requirement that the total

number of particles remains constant:

N=] flede. 0

(3.5.13)

As distinct from classical statistics, (3.5.13) is an integral equation and, in the

general case, is incapable of solution. In what follows we therefore consider limiting cases.

3.5.1 The Case of T= 0K At T = 0 the system is in its lowest-energy (ground) state. In classical statistics

all the electrons would be in a p-space cell with « = 0. However, in Fermi-Dirac statistics this is forbidden by the Pauli exclusion principle, and the electrons will fill up most densely only the p-space cells around the point ¢ = 0 so that their total energy & is minimized. With the quadratic dispersion relation (3.5.4), the

volume thus filled with electrons has the shape of a sphere with an isoenergetic spherical surface, called a Fermi surface (in the case of an arbitrary dispersion

relation the shape of the surface may differ arbitrarily from a sphere), and with maximum boundary energy @),,.,, 2 Fermi energy €;. The quantity ¢, is defined by the electron density

n=N/V.

(3.5.14)

174

3. Simple Metals: The Free Electron—Gas Model

Indeed, the volume of the Fermi sphere occupied by N electrons with two in each cell (3.5.7), because of spin degeneracy, is equal to (47/3) 3 max» WNETE Po max is the largest magnitude of the momentum

of an electron at T = OK, i.e., the

Fermi momentum pg. The quotient of this volume by the cell volume (3.5.7) is 4nVp3/3h3 = N/2 .

(3.5.15)

From (3.5.14, 15), we obtain

Pr = h(3n/8x)!3 = h(3n7n)'?

,

(3.5.16)

kp = p/h = (3n7n)'? , fp = pe /2m* = (h?/2m*)(3n/8x)?? = (h?/2m*)(3x7n)?? , or

n = (82/3)(pp/h)? = (82/3)[(2m*)*?/h? Jen?

,

(3.5.17)

and for the Fermi velocity De = pp/m* = (h/m*)(3n/8x)!? = (h/m*)(3n2n)!?

.

(3.5.18)

If we choose the value ofn to be ~ 107? cm~3, the result will be ~ 1-10 eV (or

10~'?-10~ 13 erg) for the quantity ¢, and 10° cm/s for the quantity v,. Table 3.2

summarizes the values of vp, pp, and ¢, calculated for a number of metals. As is seen from the table, the Fermi gas contains particles that possess enormous velocities and energies, even at T = 0. Table 3.2. Numerical values of the velocity v-, momentum g,, and energy & = ¢, on the Fermi surfaces of normal univalent metals Elements

vg Or fp ée

Li

[cems~'- 107} [gems~!-10'9) = Co Lerg- 1017] = Co [eV]

1.28 1.17 7.6 4.74

Na

K

Cu

1.04 0.95 5.0 3.16

0.84 0.77 3.3 2.06

1.58 1.44 11.37 7.10

Ag

Au

131 1,20 8.84 5.52

1,39 1.27 89 5.56

Our objective now is to identify the | functions n(e) and f(e) for 0K. As can be

deduced from (3.5.11), the function n(e) will depend substantially on the sign of the difference ¢—{, with C) being the value of ((7) for OK. For ¢ < Cy lim exp[(é — (9) /kg 7] = 0 and no(e) = 1;

T-0

for ¢ > fy lim exp[(¢ — €9)/kg 7] = 20 and no(e) = 0. T>-0

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

175

Thus, when ¢ = Co, the exponent [(e — {))/kg 7] changes sign and no(e) varies discontinuously from 1 to 0 (Fig. 3.6). The discontinuous no(e) curve is called the

Fermi step. Hence the chemical potential at 0 K clearly coincides with the Fermi energy

(3.5.19)

Co = fr = Eqmax -

n(€) Vy

1

dn(€) OE

Fig. 3.6. Fermi-Dirac function n(e) and its derivative Gn(e)/@e (dashed line refers to the case T = 0)

The distribution function f(¢) for T = 0, in view of (3.5.12), is equal to (Fig. 3.7)

Sole)

é)=

0,

E> ly

{anton}? Vase? , ecg.

3.5.20 (3.5.20)

f(é) 1

7=0

1

T#0

0

&

é

The mean energy of an electron at 0K +2

Fig. 3.7. Energy distribution function /(e) of electrons in a metal (dashed line refers to the case T =0)

is

i = f aeesse| f defo(e) = 30o/5 = 3e,/5 . 0

(3.5.21)

According to classical theory, this quantity is equal to é,, = 3k,7/2 and for OK &, = 0. In the classical theory, (3.5.21) could be obtained only at very high 7,

176

3. Simple Metals: The Free Electron-Gas Model

viz., at the effective gas degeneracy temperature 9,, at 0K, which is determined from the formula

fq = 3e,/5 = 3ky%ui/2

(3.5.22)

Taking the values of {, (Table 3.2) to be 107 ''-107 ! erg, we obtain

9.) = 2Lo/Skp = 104 = 105K .

(3.5.23)

Thus, due to the Pauli exclusion principle, the energy of the ground state is high, and therefore the quantum electron gas appears to be heated to very high degeneracy temperatures 9,,. For real temperatures T < 9,, the statistical behavior of conduction electrons should then be expected to deviate substantially

from the predictions of the classical theory. Only for T > 9,, should they be

expected to exhibit classical behavior. However, it follows from (3.5.37-42) that,

if the density n is close to the density of lattice sites (~ 107? cm ~ 3), 9,, exceeds not only the melting temperature of the metal but also the temperature at which it vaporizes. In all real metals the conduction electron gas is therefore highly degenerate. Only with small densities (for example, in semiconductors, where

n < 10?? cm) is the value of 9,, appreciably smaller than 10* K, and we are sometimes in a position to describe conduction electrons using classical theory.

3.5.2 The Low-Temperature Case (7 > 0K, but T < 9,,)

In the low-temperature case it is useful to introduce the small dimensionless parameter

(3.5.24)

T/9a~ keTo 0K, the n(e) and f(e) curves (Figs. 3.6, 7) will exhibit very sharp but continuous fall-

offs, called “Maxwell tails”, in the region of the discontinuity at ¢ = C9. For

T > OK it follows from a simple analysis of (3.5.11, 12) that with ¢ + oo the n(e) and f(«) always tend to zero, and when ¢ = 0 the quantity n(e) ~ 1 and f(e) +0 (if > kg 7). For « = { we have exp[(e — ¢)/kg7] = 1 and, consequently, at the inflection point of the Maxwell tail n({) = 1/2 and f(£) = g(£)/2. The quantity ¢(T) turns out to be somewhat smaller than (4, as will be shown later (3.5.51).

As T approaches 9,, the smearing should increase, and in the limit T 2 9,,

the n(e) and f(e) curves turn into classical statistics functions. ourselves to the strong degeneracy case.

We

confine

To solve particular problems, we follow Frenkel and introduce the density of thermally excited electrons present in the smeared band near the Fermi surface. Since the width of this band is related to the radius of the Fermi sphere as 7/9,,, ny = nT/9,,

.

(3.5.25)

The distribution function of excited electrons has the form of a Maxwell tail and therefore we can apply classical theory to it. We illustrate this with the example

of the heat capacity of a degenerate electron gas. To do this, we insert into the

Dulong and Petit formula

C*” = 3nk,/2, using the density n according to

(3.5.25) in place of the density n;. This yields

C@™) — 3npky = 3nkyT/29, .

(3.5.26)

As seen from the above equation, in real metals at room temperature (where the

Dulong and Petit law holds) the ratio 7/9,, ~ 0.01 to 0.001 and therefore the

contribution of electrons to the heat capacity constitutes 1.0 to 0.1% of the ionic contribution. In this fashion, the disaster with the heat capacity for a degenerate Fermi gas is completely resolved. Only electrons of density ny contribute to the heat capacity. Electrons with ¢ < ¢ — kg T cannot accept energy from the heater,

since they would enter a state already occupied, a situation which is forbidden by the Pauli exclusion principle. Returning to the more rigorous calculations, we consider the expression for

determining the chemical potential ((7) and the energy &(7) of the electron gas,

and specify how the calculation can be carried out under the condition (3.5.24).

The quantity ((7) is determined from (3.5.13):

N=C

+o

—___

f den(e)e'? .

o

(3.5.27)

178

3. Simple Metals: The Free Electron-Gas Model

Integration by parts yields

N = —(2C/3) { dee®an(e)/de .

(3.5.28)

0

For the total energy of the gas we also have

&=C va dee? n(e) = —(2C/5) ° dee*!2An(e)/de . oO

0

(3.5.29)

Thus we need to calculate a typical integral +a

1=C f dea(e)dn(e)/de ,

(3.5.30)

0

where a(e) is a continuous function ¢. For convenience, we introduce

x=(&—()/kgT

.

(3.5.31)

Then

a(e) > a(x) , and (3.5.30) becomes

1=C f dxa(x)an(xyadx .

(3.5.32)

—C/kpT

In the case of strong degeneracy, when (3.5.24) holds, the falloff of the n(x) curve

in the vicinity of x = 0 is very abrupt, and the derivative dn(x)/@x is close to a 6 function with respect to x (Fig. 3.6) and is exactly equal to it at 0K. This allows a very simple method of approximate calculation (3.5.30). Using (3.5.24), the lower limit in (3.5.32) may be replaced by — oo:

T= C f dxa(x)dn(ay/ax .

(3.5.33)

Assuming that the function «(x) near x = 0 may be expanded in a power series

ne

(x) - 10) + (2) 0x

23

x +33 0x*:) Jo x4. Jo

(3.5.34)

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

179

and substituting (3.5.34) into (3.5.33) yields

1xC —wf dx anna) 2s| 50) + (2) x+ ox 0

1/6

+3(Ga et

[ator

thee.

(3.5.35)

The first term J, in (3.5.35) can be calculated if the lower limit of the integral is

exact. This term is equal to Ig =C&(0)

+a

f

—C/keT

——

dx An(x)/dx = —CH(0)

.

(3.5.36)

The second term in (3.5.35) is equal to zero because of the function On(x)/ax is even. Indeed, by virtue of (3.5.11, 31),

8n(x)/dx = —e*/(e* +1)? = —e-*/(e“* + 1? = = 0n(—x)/0(—x)

.

(3.5.37)

The third term in (3.5.35) has the form I,=

~ C(6?&/Ax?)y

+a

j

0

dxx*e*/(e-*+ 1)? .

(3.5.38)

The integral involved in (3.5.38) has already been calculated in Sect. 2.5 (2.5.6). Thus we find

1, = —(n?/6)C(0?G/Ax")o

.

(3.5.39)

Adding (3.5.36) to (3.5.39) and proceeding from a(x) to a(), we have

lx ~c( ater

2

a2

* TA

aT)

(3.5.40)

The rapidity of convergence of (3.5.40) is determined by the closeness of the derivative

—An(e)/ de to the 6 function. The zero expansion

term (3.5.35) is

obtained if the quantity — dn(e)/de is set equal to 6(e— C) and the following terms take into account the correction for the finite “width” of the function

—On(e)/de. Subject to the strong degeneracy condition (3.5.24), all statistical parameters of the Fermi gas may be calculated to zero approximation if the distribution function n(e) is replaced by a Fermi step and — an(e)/ de is assumed

to be a 6 function.

180

3. Simple Metals: The Free Electron-Gas Model

Using (3.5.40), we calculate ((7) and &(7) according to (3.5.18, 29). In these

cases we have a(e) = e°/? and a(e) = e*/?, respectively. Substituting these expressions into (3.5.40), we find

N © 4CO?7E1 + (n?7/8)(kpT/C)?] ,

(3.5.41)

6 = ¥CC°?[1 + (Sx? /8)(kyT/C)?] .

(3.5.42)

Since kgT OK (but for kgT < 0) is

& = 3Co[1 + (Sn?/12)(kg7/Co)?J=

(3.5.46)

= éo({1 + (5n?/12)(kg7/Co)?] = eo t+ y'T?/2

,

where

(3.5.47)

y = n7kg/2lo «

The results thus obtained may be generalized for the case of an arbitrary dispersion relation ¢(p). The function g(e) will then take an arbitrary form; in place of (3.5.27-29) we will have

N= f gle)ntede = — F | face |(SO) a 0

0

oO

CE

(3.5.48)

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

& = f egle)n(e)de = — 0

0

| fateraas (5°) a de

181

(3.5.49)

0

In view of (3.5.32, 40), we obtain approximately, instead of (3.5.41, 42), §

(3.5.50)

+ (x?/6)(kaT)*g'(e) , N = fdeg(e) 0 ¢

(3.5.51)

é= J deeg(e) + (n?/6\kaT)*LCa'(C) + 9(6)] ,

with g'({) being a derivative dg(c)/de for « = ¢. Expressing ¢ in the form { = C5 + 50(|5¢| < C9) and expanding (3.5.50, 51) in df, we find, similar to (3.5.45, 46),

f= ly —(27/6)(keT)?9'(Co)/9(Lo) »

(3.5.52)

6 =Epa0 +7 r ,

(3.5.53)

where

(3.5.54)

y = 07 kgg(Co)/3 « The electronic heat capacity at constant volume is C, =08/0T=yT,

(3.5.55)

which is a more exact result than (3.5.26). Thus low-temperature measurements

of the heat capacity of metals permit a straightforward determination of the density of states at the Fermi level, g({,), which is a very important charac-

teristic. With T = 0, the energy for the free-electron mode is equal to (3.5.43, 11)

8 = 3NCo/5 = (3h2N/10m*)(3N/8xV)2?

(3.5.56)

The pressure of the Fermi gas at 7 = 0 is defined by the formula

Po = —06/OV= 28/3V =4nl, . A

numerical

estimate

with

n~ 10?2cm~3

(3.5.57) and

(,~ 107 "erg

yields

Po ~ 10'! dyn/cm? ~ 10° atm. This enormous pressure arises from the high

electron density. It is easy to understand why the electrons do not disperse

through the crystal surface, the “strength” of which is, as first pointed out by

182

3. Simple Metals: The Free Electron-Gas Model

Frenkel, of an electronic nature. A negatively charged electron, as it leaves the

metal, is subject to the attraction of the positive charge of the ionic lattice. This attractive force is approximately estimated by the Coulomb attraction of an electron to its positive image charge, see (3.1.28). At small distances from the surface this force is very large, and near the surface a high potential barrier arises which is capable of withstanding substantial electron pressure.

3.5.3 Atomic Volume, Compressibility, and Strength of Metals We wish to enlarge upon the nature of this barrier, which assures the stability of

the metal. The interaction of an electron and its image produces a negative pressure; when they are in equilibrium a balancing (equalizing) positive pressure (3.5.57) occurs. The potential energy of an electron with respect to an ion is equal to Epo = — ae? /r,

(3.5.58)

where a’ is a numerical constant ~ 1, and r is the mean distance between the electron and the nearest ion. The quantity r is calculated in terms of the electron density

rx Bn“

(3.5.59)

with the numerical coefficient B ~ 1 being determined Substitution of (3.5.59) into (3.5.58) yields

Epo = —ae?n'!®

by the type of crystal.

(a= ar'/B)

Inserting n from (3.5.14), we obtain for the potential energy

pot = NEg= —ae27NA3V~13

(3.5.60)

The negative pressure p_ is equal to p- = —06,4/8V= —aN*3e?/3V'3 = —ae?n! 3/3 .

(3.5.61)

At equilibrium, the sum of p_ and pg from (3.5.57) is equal to zero: p=p_-+po=0,

(3.5.62)

whence, from (3.5.10, 57, 61), we obtain

ry, = n"3 x (3/10)(3/82)2/2(h2/me2) ~ 1078 cm .

(3.5.63)

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

183

The above value is very close to that of the radius of the first hydrogen orbit h?/4x? me”. This elementary calculation has shown that the zero-point kinetic energy of a degenerate electron gas and the positive pressure (3.5.57) due to this energy have a reasonable dynamic meaning and give a correct estimate of the atomic volume in the metal. Now we calculate the compressibility of the metal. Note that (3.5.62), because of (3.5.57, 61) is the condition for the minimum of the total energy of the crystal:

dé d ar 7 gp (Sat Fria) =0

(3.5.64)

According to (3.5.57, 59, 60),

&=—A/r+B/r ,

(3.5.65)

where A = ae?N, B= Bh? N/2m.

If the metal is compressed or extended by an external pressure, its energy & increases in comparison with &, in the absence of a pressure. For small deformations

(3.5.66)

€— 8) 24(G78/dr?)o(r—ro)? , where r and ro stand for the crystal. The linear expansion (3.5.64), and the terms higher Express the atomic radii r

Vo=B3Nr3,

atomic radii of a term with respect than second-order and ro in terms of

V=B3Nr>,

distorted and an undistorted to (r—ro) is absent due to will be neglected. the volume of the metal:

Beni.

(3.5.67)

This gives, instead of (3.5.66),

1/a28\

6-623)

B>(V—V,)?

ee

From (3.5.64, 65) we have

(376 /dr?) = —2A/r3 + 6B/ré = Are , and, consequently,

& — &5 =(AB°/2)(V—Vo)?/9N?xr

.

(3.5.68)

The energy of an elastically deformed solid is normally represented as

& — 8&9 =(K/2)(V—Vo)?/Vo ,

(3.5.69)

184

3. Simple Metals: The Free Electron-Gas Model

with K being the bulk modulus (compressibility). Comparing (3.5.68) with

(3.5.69) and taking into consideration (3.5.64, 65), we find the statistical defini-

tion of the bulk modulus

K = AB Vo/9N?r3 = ae? N/9Voro =28/9Vo 5 where we have made

(3.5.70)

use of the virial theorem (3.4.2). From (3.5.70) we can

readily estimate the order N/Vy~(1+3)-10?2?cm~3,

of magnitude of the modulus K. Substituting e~48-10°'°CGSE, a~(1+3), we obtain

K peor © 102° + 10!! erg/cm? = 10* + 10° atm. For alkali metals experiment gives Ky, 0.6105 atm, Ky = 0.3-10° atm, and K,;~ 1.1-105 atm. Thus, the agreement between theory and experiment is quite good. Further, we estimate the strength of the metal according to Frenkel, i.e., determine the maximum

pressure p_ which the metal can withstand without undergoing fracture. The condition for the maximum has the form dp/dV =0 or, because p = —d&/dV, a’ &/dv? =0. According to (3.5.65, 67),

a6 _d (dé dr\_d[(A_2B\) 8 dV? dV\drdV)

dV\\r?

rf

|_)

/3rNI|

”

whence we find

B=2Ar,,;/5

-

(3.5.71)

In this equation r,,,, is the largest attainable radius for the atomic volume and

corresponds to the maximum possible negative pressure beyond which the metal starts to fracture (p_n.,)- By definition,

p= —d&/dV= —(B?/3r?N)(dé/dr) , or, using (3.5.65),

p= —(A/r? —2B/r>)(B3/3r?N)

.

(3.5.72)

Substituting B from (3.5.71) and A from (3.5.65) into (3.5.72), we find

P— max = — AB? /15Nr4,,= —aBre7/15r4,,

.

Setting «63/15 ~ 1, e~4.8-10~'° CGSE, and r,,,,

(3.5.73)

107° cm, we obtain the

estimated value p_,,., ~ 10'! dyn/cm? ~ 105 atm. We know from experiment that the actual strength of metals is hundreds of times smaller than this theoretical limit. This happens because uniform extension is not feasible and

overstressed regions arise inevitably in the sample. Also, as first pointed out by

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

185

Ioffe, the presence of defects on the surface, due to the concentration of stresses near them, leads to an effective lowering of the strength of the sample. Therefore, the estimate (3.5.73) may be regarded as the upper limit of the strength of metals.

3.54 Paramagnetism of a Degenerate Electron Gas All metals and alloys possess magnetic properties. Two classes are distinguished: (1) metals and alloys in which no atomic magnetic ordering (ferro-, antiferro- or

ferrimagnetic) occurs in the absence of an external magnetic field, whatever the conditions may be, and (2) metals and alloys in which this ordering is observed in a certain temperature interval. The second case usually requires that at least

one of the constituents be a transition element. Considering, at this point, normal metals, we restrict our attention to a study of the first type of metallic magnet, viz., weakly magnetic diamagnets or paramagnets.

Table 3.3 lists the magnetic susceptibilities of normal metals. As can be seen

from the table, alkali and alkaline-earth metals (except for Be) are paramagnets. Table 3.3. Atomic (x,) and specific (zy) magnetic susceptibilities of weakly magnetic (diamagnetic and paramagnetic) normal metals at T = 300 K

Element

Density [gcm~ >]

Li Na K Rb Cs Be Mg Ca Sr Ba Cu

0.534 0.9725 0.862 1.532 1,90 1.8477 1.74 1.54 2.63 3.65 8.96

Ag Au Zn

Cd

Hg Al

Ga In

a-Te

Sn white Sn gray

Pb

Bi

Xa‘ 108 +24.6 +16.1 + 21.35 + 18.2 +29.9 —9.02 + 13.25 +440 +91.2 +20.4 ~5.41

+ 1.89 +0.68 +0.47 +0,33 +0.42 — 1.83 +0.95 +17 +2.65 +0.56 —0.76

—19.7

—1.52

10.5034 19.32 TAZ

—21.56 — 29.59 —11.4

13.6902 2.70

— 33.3 + 16.7

8.65 5.91 731

11.85

7.2984

5.8466

11.3415 9.80

x 10°

—21.7 —12.6

21 -29 —1.24

—2.25 + 1.67

— 1.84 —08

— 58.0

—3.37

-37

—0.184

+45

— 24.86

— 284.0

+0.276 — 1.36

— 13.0

186

3. Simple Metals: The Free Electron-Gas Model

Their susceptibility is low (jm ~ 10~°) and is practically independent of temperature. Of the group-III and group-IV elements, only aluminium and white tin are paramagnets. The other normal metals are diamagnets which (except for Bi) exhibit low susceptibility (|Zam| ~~ 107°) and 74.,/dT= 0. Several questions arise here: Why are some of the metals paramagnets, and

some of them diamagnets, and why is their susceptibility practically independent of temperature? A free electron, possessing an intrinsic mechanical moment (i.e., a spin) and an electric charge, should also possess a magnetic moment p,,. Relativistic mechanics tells us that the spin-magnetic moment of a free electron is equal to the Bohr magneton

Hy =|elh/2me . If conduction

(3.5.74)

electrons obeyed

the laws of classical statistics, their para-

magnetism would be similar to that of normal gases. Specifically, the tempera-

ture dependence of the susceptibility of a gas of density n would obey the Curie law

(3.5.75)

Apm(T) = npp/kyT .

The factor of 3 is absent in the denominator of (3.5.75) because the spin-magnetic moment may have only two projections relative to the field

rather than any projection, as is assumed in the classical theory. In explaining the magnetic properties of normal metals, we must take into account that the “source” of magnetism is the ion core lattice and itinerant electrons (the contribution of nuclear magnetic moments may be neglected because of their smallness). The simplest case is that of the alkali metals, in which

the electronic shells of ion cores are identical to the closed shells of the atoms of inert gases possessing weak diamagnetism. Therefore, the paramagnetism that

occurs in these metals is due to the paramagnetism of the conduction electrons, ie. 49 = X$m — | in|. The same is true of the alkali-earth metals (except for Be),

Al, and £-Sn. In all other cases |zi93|> Sq, and therefore these metals are diamagnetic (in reality, the diamagnetism may sometimes arise also from

conduction electrons; for Bi, e.g., see Sect. 3.5.5). The first to pay attention to this circumstance was Dorfmann [3.6] who also pointed out another fact: if one

compares the observed atomic susceptibility of normal diamagnetic metals with the susceptibility of their ions (obtained, for example, from salts or salt solutions containing these ions), the quantity | 72*'| is always smaller than |z'°*|. Thus, for copper, silver, and gold we have Element

ni, 10° yan. 10°

Cu

-541 -180

Ag

Au

-2156

—29.59

-310

-458

3.5

Application of Fermi~Dirac Quantum Statistics to the Conduction-Electron Gas

187

Proceeding from this fact, Dorfmann inferred that conduction electrons possessed paramagnetism and it was this paramagnetism that decreased the diamagnetic susceptibility in going from ions to metals. Since yfs', as a rule, is independent of 7, the paramagnetism of conduction electrons, according to Dorfmann, must also be independent of 7. Thus the absence of a dependence such as (3.5.75) for all normal paramagnetic and diamagnetic metals turned out

to be one of the most conclusive proofs of the inability of classical physics to explain the properties of electrons in a metal. Thus the papers concerned with the explanation of the temperature-independent paramagnetism and diamagnetism of normal metals [3.6-9] may have marked the beginning of not only the quantum theory of the magnetic properties of these materials, but also of quantum (electronic) solid-state theory as a whole. We start with the simplest case of the paramagnetism of a Fermi electron

gas. This paramagnetism occurs if x51, > |z'fs| and the diamagnetism of the

electrons themselves may be neglected. These requirements are met best in the alkali metals. In the absence of an externally applied magnetic field the total magnetic

moment of the Fermi gas at 0K is equal to zero because of the complete compensation of electronic spins. This shows that the Pauli exclusion principle in a system of fermions leads to a substantial dependence of their energy on the magnetic moment, even if the magnetic forces are disregarded. As will be seen later, this dependence appears even when we take into account the electrostatic interaction of electrons, thus permitting an explanation of the ferro- and

antiferromagnetism In magnetizing move the electrons energetic influence

H ~10* Oe,

it

fo ~ 107 !? erg.

(Chap. 5). a highly degenerate electron gas, the magnetic field has to from states with energy &(Co. The of the field is determined by the quantity gH; with fields

will

amount

to

~107!*erg,

which

is

much

less

than

Therefore, in the main, the redistribution of electrons by the action of a field

occurs in a thermal-smearing band of width ~kg7 near the Fermi level. To

estimate the paramagnetic susceptibility, one may employ, according to Frenkel, the classical formula (3.5.74). However, for thermally excited electrons, (3.5.25)

implies

ila = np ua/kyT= np2/kp Sey -

(3.5.76)

In this approximation the temperature independence of the paramagnetism of

an electron gas is explained immediately. In addition, we obtain from (3.5.76) a

numerical estimate of y%,,~ 10~®=+ 10-7

which

tallies with the data sum-

marized in Table 3.3. Only at very high temperatures (72 9,,) will the electron gas behave as a classical one, obeying (3.5.76). Since 9%,, is higher than the

vaporization temperature of metals, the classical paramagnetism is virtually unobservable in normal metals. Exceptions occur only for a few poor metals

188

3. Simple Metals: The Free Electron—-Gas Model

which have a low conduction electron density and a low 9,,. Let it be emphasized that what is meant here by normal metals is solely nontransition metals. Let us now consider the changes which the Fermi distribution undergoes by the action of a field. With H=0 and with the dispersion relation being

quadratic, the density of states g(e) has the form of a parabola. The density of

states may be introduced separately for electrons with “right-hand” and “lefthand” spin projections (sometimes one also talks of spins directed “upward” or “downward”) g,(e) and parabolas (Fig. 3.9a)

g_(e); these functions

also will have

the shape

of

(3.5.77)

g+(e)=g_(e) =49(e) . Aed

é-aL--4

F

HO

&

m "

3.5

6-0

Fig. 3.11. Landau levels arising from the continuous spectrum

&€ =Cy. When H # 0, the band splits into strips of width Ae = 2ug A[(l + 1) + 1/2 -—1-1/2] =2pygH

,

(3.5.104)

each of which converts to one discrete Landau level that lies exactly in the middle of the strip (Fig. 3.11). The degree of degeneracy of each discrete level will be proportional

to 2ug,H.

The emergence

of Landau

levels for conduction

electrons in a magnetic field may also be represented in p space. Figure 3.12 depicts a Fermi sphere octant ACB. The occurrence of Landau levels boils down to replacement of the continuous sphere by a set of inscribed concentric cylinders that have a common axis p, and are spaced a distance of 2u,H apart. To determine the diamagnetic susceptibility of an electron gas, we have to calculate the partition function Z(7, H) and then determine the thermodynamic

potential $(7, H) = —k,7InZ and the magnetization

1 = —0$/0H =kgTOInZ(T,H)/OH .

(3.5.105)

For simplicity, we start by considering the Maxwell statistics case. Then

Z(T, H)= Ymexp(—e/kp T).

(3.5.106)

Fig. 3.12. Manifestation of Landau levels in momentum space

196

3. Simple Metals: The Free Electron-Gas Model

At the outset, we find the values of the statistical weight g, for particles with the

energy spectrum (3.5.103), ie., the number of states in the volume of a phase

space as a cylindrical ring of height dp,, internal radius p, = (p2 + p?)'/?, and width dp, (Fig. 3.13). The volume of the ring is equal to 2 p, dp, dp,. According to (3.5.103), in quantum mechanics

pi/2m — & — p2/2m = 2p, H(I + 1/2) , P.dp, > 2mugH Al = 2mpgH

,

Hz

C

Py

dP,

ah, Fig. 3.13. Calculation of the statistical weight for electrons in a magnetic field

since Al = 1. If we recall that the size of a unit cell according to (3.5.7) is equal to h? /V and if we take account of the spin degeneracy, the result for g,, obtained by

replacing the ug, according to (3.5.74), will be

9, = 2np, dp, dp, 2V/h? = 2n- 2mpy H(2V/h>) dp, = 2\e|VHdp,/h'c

.

(3.5.107)

Substitution of (3.5.103, 107) into (3.5.106) yields the following [here we intro-

duce a new variable x? = p?/2mk, T and use the conditions [dx exp(—x?) =n"?

-2D

and +2

> exp[—(2)+ Iugh/kg TJ] =e "(l+e° %+e

1=0

+...)

=e %(1—e77%)"1 =(2shy)7'

,

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

197

where y = up H/kgT]:

Z(T, H)= ¥ 7 dp, 2\e| VH/h?c i=0

-a@

x exp { — [pp H(2I+ 1) + p2/2m](kgT)~'}

(3.5.108)

= (le| VH/h?c)(2amky T)"? [sh(ug H/kyT)]~! = (2nmky T/h?)"? (up H/kyT)[sh(HpH/kyT)]~* . Using (3.5.105), we obtain for the magnetization I= NkgToélnZ/0H = — Nug(cth(ugH/kyT) — kg T/p gH) = — NypLl(upH/kgT)

,

(3.5.109)

with L(x) =,cthx — x7! being a Langevin function. For weak magnetic fields

and not very low temperatures—i.e., subject to the condition that zyH < k, T—

(3.5.109) simplifies to L(x) x x~! + x/3 —x3/45+ for the susceptibility the Curie law

Xam = — np /3k_T

...—x71, and we obtain

.

(3.5.110)

It follows from (3.5.110) that in the “classical” case the diamagnetic susceptibility of an ideal electron gas is equal in absolute magnitude to one third the paramagnetic susceptibility: see (3.5.75) or (3.5.90).

In the derivation of (3.5.110), we assumed that, although possessing a quantum spectrum, the electron gas obeys classical statistics, whereas actually it obeys Fermi-Dirac statistics. However, we can find from (3.5.110) the correct

expression for y{!, without performing, again, a statistical calculation but, using the already familiar Frenkel method, by replacing the total density n in (3.5.110) by its thermally excited part (3.5.25). If, in addition, we allow for the factor 3/2 [transformation from (3.5.76 to 80)], the result for the diamagnetic susceptibility

will be [3.9]

xd, = — y/o = — (4mpZ/h?)(n/3)?9n'? .

(3.5.11)

Let us now look at a more rigorous derivation of (3.5.111). To this end, we must find the thermodynamic potential (7, {) of an ideal Fermi gas with a spectrum ¢,. This potential can be readily found using the distribution function (3.5.11) and the thermodynamic relation

N =D nie) = ¥ {exp[(e,— 0)/kaT] + 1}7! = — 00/0 .

(3.5.112)

198

3. Simple Metals: The Free Electron-Gas Model

Integration with respect to ¢ yields

Q = —kgTYIn{1 + exp[(C—e,)/kaTI} +C ,

(3.5.113)

where C is independent of ¢. For (/kg T— 00 we should have Q=&—CN = Yue< to (Ev — Co), whence we find C = 0. Inserting into (3.5.113) the Landau spectrum (3.5.101) in place of ¢, and setting V = 1, ie. N =n, we obtain +0

+0

Q = —(82m/h?) pg H kp Ty J dp, In(1

(3.5.114)

+ exp{[¢ — 2p, H(I + 1/2) — p2/2m](kyT)~"}) . Here we have employed

(3.5.107) for g, per unit volume. The summation

in

(3.5.114) may be carried out with the help of the Euler formula [3.13], which, to within the approximations adopted, has the form

i) + Sa).

Liu 1/2) =fax so)

(3.5.115)

Formula (3.5.115) is applicable provided that the function f(x) is close to a linear

one in the interval (a, b) between two values of the argument x that differ by

unity, i.e, under the condition

[f(x+ 1/2) — f(x — 1/2)-f'()1 < IF) The integrand in (3.5.114) varies appreciably in the interval for |, which is equal to ky T/U, H at some points of the entire interval of /, i... 0 < | < oo. Therefore, the Euler formula (3.5.115) can only be used for calculating (3.5.114) provided that

Hp H/kpT { dp, § dxin{1 + exp[(¢—2upH, -x

0

— p2/2m) (kg T)~* ]} + (2am/3h?)(ugH)?

x f dp,{1 —exp[(p2/2m —O)(ke TV}?

(3.5.117)

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

199

Proceeding to the new variables (first 24, Hx = y, then p2/2m = z, and, finally,

y+z=n) and changing the order of integration with respect to z and 7, we obtain, instead of (3.5.117),

Q = —[4n(2m)"?/h?] ky Tf don? In {1 oO

+ exp[( —1)/ky TJ} +[2(2m)?2/3h?](u,H)? x fai7/(

t+exp[(z—()/kaT]}"'

(3.5.118)

.

Applying to the integrals in (3.5.118) the procedure employed to calculate the integral (3.5.32), we find for the zero approximation Q = —16n(2m)>? 65/2/15h> + [2n(2m)?/?0'/?/3h3](p_pH)? For the magnetization we obtain

we have I = —(6Q/éH),.

.

(3.5.119)

Using (3.5.119) with ¢ = Co,

[= —[4n(2m)?2/3h3] 87? p2H Replacing (, according to (3.5.19), we find for the diamagnetic susceptibility according to Landau

Gm = — (4nm/3h?)(3n/x)"? ps ,

(3.5.120)

which coincides exactly with (3.5.111). Using in (3.5.119) expansions in higher

powers of pt, H,,, it is possible to calculate the dependence of yé!,, on T and H for

strong degeneracy [3.14]. However, the result obtained is valid only for free

electrons with a quadratic dispersion relation and when (3.5.115) holds. One important remark must be made. When we wrote the quadratic

dispersion relation (3.5.4), we labeled the conduction-electron mass m* because

it may differ from the mass of a free electron in vacuum. On the other hand, the

Bohr magneton pg involved in the formulas for y5!, has appeared as the replacement of a definite group of constants entering into the expression of wy according to (3.5.92) [transformation from (3.5.101) to (3.5.103)]. Therefore, strictly speaking, (3.5.120) should be rewritten as

Xam = — (4m* y5/h?) (2/3)? (m/m*)?

(3.5.121)

[one factor m* arises from (3.5.19) for Co]. Thus the total susceptibility $1, of an

electron gas, when the dispersion relation is quadratic but the mass is only the effective mass, will have the form

200

3. Simple Metals: The Free Electron—-Gas Model

.

;

;

Ato = X$m + Xm =

12m*2

re

(n\2/3

*(5)

1f/m\2

1-375)

|-

(3.5.122)

Equation (3.5.122) demonstrates that yf, = 2y$1,/3 only when m* = m. But if m* > m/./3, the electron gas is always paramagnetic overall. Conversely, for

m* < m/./3 it is always diamagnetic. Thus, Landau diamagnetism is fundamentally observable in a gas of electrons with small effective masses

(m* < m/,/3).

3.5.6 Oscillatory Effects in the Fermi Gas In the general case the energy of an electron in a magnetic field is the sum of the

energies (3.5.78, 103):

e(l, p,, H) = p2/2m* + hoy (21+ 1) — payHH .

(3.5.123)

Substituting (3.5.123) into (3.5.113), we can, in principle, calculate the magnetic

susceptibility of an electron gas without dividing it into a diamagnetic and a paramagnetic part, as was done in the foregoing, and also take more rigorously

into account all the features peculiar to Fermi-—Dirac statistics. These calcu-

lations are very cumbersome and for this reason we restrict our attention to an

elementary method of ascertaining some peculiarities of the magnetic properties of the Fermi gas.

As far back as 1931 De Haas and van Alphen discovered experimentally that the magnetic susceptibility in bismuth varied in a periodic fashion when the magnetic field was varied in the region of low temperatures. This phenomenon may be explained if we give up the restriction imposed by the condition

gH

H,

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

201

for which the number of Landau levels with energies smaller than or equal to is equal to N’ or N’ + 1, respectively. Then

Co/2MaH,=N’,

(o/2ugH,=N'+1,

whence we find for the oscillation period A(1/H) = 1/H, —1/H, =2 up/fo -

(3.5.124)

Peierls [3.15] investigated the magnetism of electrons subject to the condition

MaH > kyT .

(3.5.125)

He considered a two-dimensional electron gas at T= 0

K in a magnetic field

perpendicular to the plane of the field. In this situation the energy levels are, according to (3.5.123), equal to

&, = fy (21+ 1) . In the two-dimensional case the degree of degeneracy, according to (3.5.107), is

$= 2le|HS/ch= BH,

B=2elS/ch,

(3.5.126)

with S being the area of the system. If g, is larger than the total number of electrons N, all of them occupy the state with |! = 0 and the total energy & is — NppH. According to (3.5.107), the magnetization | = ng and zero suscepti-

bility correspond to this energy. As the magnetic field H is decreased, the total

energy & decreases until the g, become smaller than N. According to the Pauli

exclusion principle, some of the electrons will then pass to levels with | = 1 and,

in consequence, with decreasing field the energy increases; i.e., the system becomes paramagnetic. It is easy to find the general expression for the total energy &, when N electrons completely fill r lowest-lying Landau levels and

partially fill the r + 1 level, according to the inequalities rBH Co is vanishingly small. Our object now is to calculate the thermionic current, which is equal to the number of electrons that evaporate from unit surface of the metal per unit time. The dispersion relation is assumed to be quadratic. Since the potential (3.5.130)

depends only on x, only the momentum component p, is of interest to us. The energy of the electron which can escape from the metal through the surface (y, z)

should

exceed

the sum

(p; + pz)/2m*, ie.,

of the potential

barrier energy

c> W+(p2 + p2)/2m* = Co + w+ (p? + p2)/2m* =e, .

and

the quantity

(3.5.132)

The total number of electrons per square centimeter per second with momentum between p, and p, + dp, that penetrate the metal is equal to

a

3.5

Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas

2_

2.

de

pail Pas Pyr p)v, 4p, dp, dp, = pt (Pxs Py» Pe) ap, dp, dp,dp, .

205

(3.5.133)

The flux of electrons can then be found by integrating (3.5.133) over all values of

Px, Py, and

p, that satisfy (3.5.132). As we do so, we take an integral

de (de = (0e/0p,) dp,). This gives 2

to

over

+a

ia Sf dp,ap, § de{1 +expl(e—0)/ky7]}' =

aT

(3.5.134)

Tf dp, dp, tnt +exp{—(kg7)"!

—a

x [w + (py + pz)/2m*]}) . Here do not use the distribution function that gives the number of electrons in

volume V with energies ranging from ¢ to ¢ + de—i.e., the function from (3.5.12)—but rather the function y(p,, p,, Pz), which gives the number of electrons per unit volume with momentum components from p, to p, + dp,, etc, and which is equal to (2/h>)x A(p,, py, p,). (In taking an integral over de we replace the variables exp[(e—¢)/k, 7] =u. The replacement yields

kp TS, © [u(u + 1)]~1-du, this integral being easy to take.) Under normal conditions w > k,T (it is known from experiment that w ~ 1 + 10 eV, and at normal temperatures kg T ~ 0.01 eV); therefore,

In([1 + exp{ —(kg 7)"

'Lw + (p? + p?)/2m*]})

= exp{ —(ky T)"*[w + (p? + p2)/2m*]} . Upon integration over dp, and dp,, we find for the thermionic current

jin = (4nem* ky /h?) T? exp(—w/kpT) = AT2exp(—w/kgT).

—_(3.5.135)

The factor exp(—w/k,7) plays the main role in (3.5.135); in comparison with this factor, the term T? is practically unobservable experimentally. In the

classical theory (Richardson) the result obtained instead of (3.5.135) was

joi = en(ky T/2nm)'? exp(— W/kgT)

,

(3.5.136)

derived in the same way as (3.5.135) if in place of the Fermi distribution one

substitutes into (3.5.135) the classical function exp[(¢., — €)/kg 7]. It is nearly

impossible to perceive

the difference between

the factors

T?

and

T’?

in

(3.5.135, 136) in the presence of the factors exp(—w/kg T) or exp(— W/kgT).

However, both formulas differ substantially in that in the quantum formula the

206

3. Simple Metals: The Free Electron-Gas Model

exponent involves the quantity — w/kg T, and in the classical formula it involves the quantity — W/k,T; i.e., they involve different work functions.

The decrease of the work function in the quantum case may be qualitatively explained as follows: the Fermi energy ¢, entails an internal electron pressure

(3.5.57) p ~ nC which facilitates the penetration of electrons through the potential barrier (“squeezes out” the electrons). Let it be also noted that

in

the

quantum

(A = 120 A/cm? K2)

formula

(3.5.135)

the

factor

A

has

a

universal

value

for all metals, whereas in the classical formula (3.5. 136)

it

involves n—a quantity that is individual for each particular metal. Table 3.4 shows that the experimental values of A coincide with the theoretical ones only in order of magnitude. This is due to the crudeness of the model used in the

derivation of (3.5.135) [Ref. 3.6.6, Sect. 30).

Table 3.4. Thermoelectric constants A of some metals [3.16]

Metal Ca

Cs Mo

Ni

A (A/cm? K?] 00

160 60

27

Metal Pt

Ta Th

Ww

A[A/cm? K?] 10*

50 60

60

3.6 Transport Phenomena 3.6.1 The Boltzmann Kinetic Equation Conduction electrons in a metal may be subject to an electric fields, a magnetic

field, and a temperature gradient. In addition, they suffer collisions with each other, the ionic lattice, its various defects, etc., and a dynamic equilibrium

results. In this situation, the electrons lose the energy and momentum they have received from the field in scattering processes. To study a set of transport phenomena (electrical conductivity, thermal conductivity, Hall effect, etc.) we use the Boltzmann kinetic equation. Now we concentrate on its derivation. The state of the system is statistically described with the help of a distribution function in the phase space of coordinates and velocities, which may also

depend on time t:7i(r, v, t). The number of electrons in a volume element of the

six-dimensional phase space is equal to dr, dt,. A(r, v, t)dt,dt, .

3.6

Transport Phenomena

207

The normalization conditions for the function n are chosen as

fat, dz,Alr, 0, t)=N

,

(3.6.1)

where fi is integrated over the entire phase space. To find a kinetic equation satisfied by the functions 7i(r, v, t) in the presence of externally applied fields and scattering processes, we single out a volume element in the phase space and consider all possible variations of these functions in the volume element. The quantity 7 can vary with time for a number of reasons: first, there is the explicit i dependence on t, determined by the partial derivative of with respect to time: dn(r, v, t)/Ot; second, the diffusive variation of ni owing to the transport of particles from one segment of r space to another and a field-induced variation of # owing to the acceleration a = (a,, a,, a,) in external fields. Thus, those particles which at a time t were in a cell with coordinates x—v,dt, y—v,dt, z—v,dt; v,—a,dt, v,—a,dt, v,—a,dt will

emerge at a time ¢ + dt in a given phase-space cell with the mean coordinates

X, Y, Z} Vy, Vy, V,. This will be sufficiently exact if the time interval dt is so small

(dt 1), the quantity 4@,,/Qp tends to saturation ~ B/C.

Let us ascertain the meaning of weak and strong fields in this case and the physical cause of saturation. The boundary between these fields may con-

=? f

Fig. 3.20. Dependence of magnetoresistance on external magnetic field

228

3. Simple Metals: The Free Electron—Gas Model

ventionally be determined from the equality CH? = 1. According to (3.6.63), this signifies that the radius of the electron orbit is equal to the mean free path:

ry=l. Thus, in a weak magnetic field | ry, the electron may make several turns in the cyclotron orbit in the magnetic field before it suffers a

collision. In strong fields the resistivity therefore does not build up but reaches saturation. For some time, the most thorough magnetoresistance measurements had

been carried out by Kapitsa in 1929 [3.23]. His experiments showed that in low fields (3.6.87) gives a good fit to experimental data. In high fields the situation is

much more complicated. First of all, normal metals possessing good conductivity exhibited no saturation up to the highest fields attained by Kapitsa (~ 3-105 Oe = 2.4: 107 A/m). Saturation was detected only in the case of poor metals (bismuth) and semiconductors (germanium, silicon). Secondly, the numerical value of the coefficient B, calculated according to (3.6.88), turned out

to be approximately a factor of 10* lower than the observed value. In addition,

most of the metals were found to display a large region of linear dependence of

magnetoresistance on external field—this is the Kapitsa law. In some cases the linear trend turned out to be intermediate between two

quadratic dependences, or occurring before the transition to saturation, but sometimes deviations from the linear trend could not be observed even in very

high fields of several hundred kilooersteds (dotted line in Fig. 3.20).

Finally, it was found experimentally that the electrical resistivity varies in a longitudinal magnetic field (H,, 4 0). According to the electron gas model, in both the classical and the quantum version, such an effect should not take place. As stated above, a magnetic field is incapable of introducing asymmetry into the

distribution function along the field direction, since it does not affect the parallel component of the electron velocity. All these difficulties are associated with the major simplifying assumptions of the Fermi free-electron gas model, which employs an isotropic quadratic dispersion relation. It may be expected that these difficulties will be resolved only in a more rigorous theory (Chap. 4).

3.7 High-Frequency Properties 3.7.1

Basic Equations

One of the characteristic properties of metals is their luster. In every day life we often distinguish metals from nonmetals by metallic luster. Of great importance

in electrical engineering is the so-called skin effect (nonpenetration of an ac electromagnetic

field into

a metal).

Finally, a vast and

rapidly

developing

3.7

High-Frequency Properties

229

domain of solid-state physics is the study of the various types of electromagnetic waves in metals and semiconductors. Many of these phenomena, relating to the propagation of an ac electromagnetic field in semiconductors, may be understood satisfactorily in terms of the free-electron gas model. We should start from a system of Maxwell equations

divb=0,

(3.7.1)

div D=4nQp

,

(3.7.2)

106 curl E ey Be co

cde (3.7.3)

10D

curl A = ca

4n,

+ Tho -

(3.7.4)

The quantities b and D here stand for the magnetic and electric induction; & and E are the strengths of an ac magnetic field and an electric field, respectively; Q9 and j, are external-charge and current densities, which will be assumed to be

equal to zero. For the system of equations (3.7.1—4) to be of closed form, we need

also to set down the so-called material equations relating D and h to E and b:

D=E+4nP,

(3.7.5)

h=b-4aM .

(3.7.6)

The polarization P is the field-induced mean dipole moment of unit volume

P=

(Fen)

5

where the summation is taken over all i particles with x; coordinates and e; charges; the angular brackets denote averaging over a nonequilibrium distribution in a field. The quantity P is related to the induced current density |

i= ne

dx;

a)

by the equation

OP/at =j .

(3.7.7)

The magnetization M is usually small in normal metals because of the smallness of the susceptibility y. In spite of this, a number of high-frequency phenomena

230

3. Simple Metals: The Free Electron—Gas Model

exist in which it plays a decisive role (e.g., electron-spin resonance). We will not

consider such phenomena here, setting 6 = A.

We will be concerned with a monochromatic electromagnetic field of frequency w with h, E ~ exp(— it), 0/dt + — iw. Taking account of what has been stated above, we transcribe (3.7.14) into

divh=0,

(3.7.8)

divD=0,

(3.7.9)

curl E= on ,

(3.7.10)

curl h = - (E+ c7'(v k))- Ofi/dv, in addition to the replacement of (3.7.14). Nothing, however, altered since (vx hyo” = m*(v x A): ov

is in, x is

fig =0. de

The wave field & should not be confused with the external magnetic field H, which, in keeping with Sect. 3.6.5, enters into the kinetic equation as the terms

~ Hn,; allowance for the terms ~ hn, would be an excess of accuracy since h~E and fn, ~ E. Thus, this point needs no alteration in comparison with Sect. 3.6. Finally—what is most important—an ac electric field is necessarily spatially

inhomogeneous. That is to say, fi, will also be inhomogeneous, and in the kinetic equation we have to allow for the diffusion term (v V,)f,, which was not taken into account in Sect. 3.6. Allowance for this term leads to what is known as nonlocality (spatial dispersion) effects, which we will consider in Chap. 5 when we discuss the plasma model of a metal. Here the diffusion term will be neglected, as it is small compared with 07, /Ot or b, —b_, provided

v/6 < max(w,77') ,

(3.7.16)

where v is the characteristic electron velocity (in metals this is the Fermi velocity), and 6 is the reference length over which the field (and, consequently, n,) varies substantially. Assuming (3.7.16) to be fulfilled (local regime), the current j is expressed in terms of the field E by the formulas of Sects 3.6.2 and 3.6.5 with the replacement of (3.7.15).

3.7.2 Skin Effect To start with, we consider the penetration of an electromagnetic field into a metal in the absence of a dc magnetic field. Then, according to (3.7.15), (3.6.30) 2

j= =i

— iwi)" E .

(3.7.17)

Substitution of (3.7.17) into (3.7.14) yields

Vdiv E—- 4E =a

cea)E ,

(3.7.18)

232

3. Simple Metals: The Free Electron-Gas Model

where the notation introduced is e(w)= 1 + 4nine?z/mo(1 —iwt) = 1 + itw2/@(1 —iwt)

(3.7.19)

and &(w) is the dielectric constant of the metal at a frequency w. The quantity

@, = (4nne?/m)'/?

(3.7.20)

of dimension s~! is called the plasma frequency; its meaning will be clarified

later on. For good metals w, ~ 10'®s~'.

Let the metal occupy a semispace x > 0 and let the electromagnetic wave be normally incident at its surface (a generalization for the case of inclined

incidence presents no difficulties). Then E depends only on x. The electric field vector should be normal to the propagation direction, i.e., E, = 0. Then

div E=0E,/dy + 0E,/dz=0,

AE=07E/dx?

,

and (3.7.18) takes the form

6? E/dx? —(w?/c?)e(w) E=0 «

(3.7.21)

The solutions to (3.7.21) behave differently for different w. At the outset we explore the low-frequency region

wt 1,

(3.7.46)

the quantity (3.7.45) and, consequently, the absorbed power (3.7.46) exhibit, as a function of frequency (or field), a sharp maximum for a positive circular component when

@ = |wy| = |e| H/me

(3.7.47)

(recall that e 1. Therefore, the energy spectrum for E > Vg also

has the shape of continuous bands separated by forbidden gaps. This part of the spectrum is depicted as shaded bands in Fig. 4.3b. A comparison with Fig. 4.3a

shows that the periodicity (with E > V,) splits up the continuous spectrum of a single well into bands separated by gaps, and for E < Vg it splits up the isolated

levels into continuous bands that are also separated by gaps.

Determine the position and width of the gaps for E > V,. From (4.1.30) we

find, up to the quantities ~ € = V,/E, —1 2. Finally, the third type of solutions results from the second type with € = 0, 7; ie., it corresponds to | f(E)| = 2. Therefore, if E is such that |f(E)| < 2, there are two linearly independent complex conjugate solutions of (4.1.12), which are given by (4.76-79). Any other solution may be represented as a linear combination of these two. If the energy is such that | f(E)| > 2, there are two linearly independent real solutions of (4.1.12), determinable from (4.1.80), which, however, should be rejected as unnormalizable. Finally, if | f(E)| = 2, one

solution exists, having the properties of (4.1.82).

264

4. Band Theory

Thus the structure of the energy spectrum of the array is wholly determined

by the form of the function f(E), plotted in Fig. 4.8 (see Fig. 4.2a, b). The domain

enclosed between the straight lines parallel to the abscissa axis at f(E) = +2 is called the internal region, and the rest the external region. Every E for which J (E) lies in the external region is forbidden, and the E for which f(E) lies in the internal region (intervals E, — E,, E, — E3, Eg — Es, and E, — E;) is an allowed and two-fold degenerate energy value; finally, the E for which f(E) lies at the boundary between the regions [at f(E) = + 2] corresponds to a special case (see

below). Figure 4.8 shows that this case occurs for discrete E values. f(E) |

\

T I

ey

{

LN 1

'

1

1

'

i

1

toa

Ap oN

'

fe fs

4 &

A

T

t

1

'

ft

'

\

foe

eg

AN

0

—~E

Fig. 48. Function f(E) from (4.1.83)

To find the energy spectrum, we consider two limiting cases. One of them is when negative energies are very large in magnitude; i.e., when (4.1.12) at finite V (x) takes the form

ae RW2 =0,

dy

2 k= —2mE/h?

(4.1.86)

2

Solutions to (4.1.86) will be y,,.(x) =e*™. Comparing them with (4.1.81), we see that starting from some point, a large negative E is bound to become unallowable. Equation (4.1.85) shows that in this situation f(E) = 2cosh kd > + 00, for k > oo when E > — oo. Thus the extreme left-hand branch of f(E) has

a shape such as that portrayed in Fig. 4.8. If E> 0 and is very large, then 2mE/h? =k? and k— oo, and the wave functions have the form of plane Therefore, with E — + 00, we are of allowed solutions, which in the This is clear, for when E > | V(x)|,

waves: W, (x) = exp(ikx), W2(x) = exp(—ikx). certain, sooner or later, to get into the region limit transform to free-electron plane waves. the inhomogeneities of V(x) cease to affect the

electron motion. Thus, as it goes over to the right-hand extremity of Fig. 4.8, the

J(E) =2coskd

curve

becomes

periodic,

being

confined

within

the

internal

region and touching the boundary straight lines at the extreme points only. In

the intermediate region of finite E the f(E) curve is always finite for, as (4.1.85) shows, with f(E) > oo and & — oo both functions (4.1.80) would go to infinity in

the entire semispace, which is impossible. In all cases except for E + — oo, the

4.1

Preliminary Observations and the One-Dimensional Model

265

J (E) curve is therefore finite and regular. Figure 4.8 reproduces schematically the oscillatory trend of the f(E) curve (see Fig. 4.2). As is seen from the figure,

the spectrum of an electron in a one-dimensional array consists of continuous allowed energy bands (heavy-line segments on the E axis) separated

by for-

bidden energy gaps (thin-line segments on the E axis). This is similar to the results of model problems (Sects. 4.1.2, 3). Making use of the f(E) curve, we will determine the numeration of the wave functions (4.1.78). In the internal region we break this curve up into branches between adjacent maxima and minima; if these extrema lie in the external region, what we will understand by branches will be segments of f (E) between the point

of its “entry” into the internal region and the point of “exit” closest to it. To each

branch corresponds a determinate set of wave functions, called a band. Thus, the

steady states of the electron in a one-dimensional array are subdivided into classes that form a discrete sequence of bands. To

each

interior

point

f(E)

corresponds

a pair

of functions

(4.1.78)

defined by the band number ¢ and the value of the function {(E). To specify

each of the functions, we make use of the quantity € from (4.1.76) or (4.1.86):

&€=-—

iln dA, =iln A, =cos~'[f(E)/2]. For instance, to the first of the func-

tions (4.1.78) we ascribe the value € = —iln4,, and the second —&é = —iln A,, differing in sign. Each wave function then is defined by the band number { and quantity €. However, as is manifest from (4.1.76), € is ambiguous and is determined to within 22n (with n an arbitrary integer). To lift this ambiguity, we

henceforth consider a € that satisfies the inequality —na aay — ee), — R5BLRS — KS) x [E(k'C’)— E(k’E")] j drufy(r)

(4.2.39)

Vo

XVyetipe(r),

C#C".

We conclude this section by clearing up the question of the electron current

at the boundaries of a zone. In the interior of a zone the wave functions can be arranged in different ways and, depending on the mode of arrangement, the

states ascribed to a zone boundary will be different. Having no opportunity to enlarge upon this problem, we restrict ourselves to the following postulate: For a

number of crystals, by virtue of their specific symmetry, the current vector in eigenstates is normal (orthogonal) to certain planes of & space. Whatever the zone construction method, planes [Ref. 4.5, Sect. 17].

these planes normally

turn out to be boundary

4.2.4 The Properties of Constant Energy Surfaces Thus far we have used only translational crystal symmetry. However, we can also consider other symmetry elements—proper and improper rotations and

reflections that form the point group of the crystal, Q (Sect. 1.2). Then, just as in the case of (4.2.1), we have

(4.2.40)

QV(r) = Vir) .

Each of these operations can be compared with the coordinate transformation x=Yajx,, I where the conditions y

3x3

ij Qy = Sy -

if =1,2,3, matrix

elements

(4.2.41) of the

transformations

||a;j||

satisfy

the

(4.2.42)

The Schrédinger equation (4.2.17) in this case will not be invariant under the transformation (4.2.41) because of the term 2i(k- V)u,,(r). To assure invariance,

we need simultaneously to transform the wave vector k as well. (We leave the proof for the reader; see Jones [4.5].) Thus

i= Dak;

(4.2.43)

290

4. Band Theory

It then follows from the invariance (4.2.17) that in each band ( E(k, {) = E(k’, fC)

or

(4.2.44)

QE(k, C) = E(k, ¢) ;

i.e., in each band the energy E(&, {) as a function of k possesses the complete point symmetry of the crystal. Let us now ascertain some general properties of the constant energy surfaces (including Fermi surfaces) associated with crystal symmetry. In particular, just as for the function E(k,¢), the surfaces E(k,¢)=const should possess the

complete symmetry of the point group of the crystal. Also note that there is a

direct correlation between the transformation in r space (direct lattice) and the transformation in k space (reciprocal lattice); namely, if the direct lattice possesses symmetry axes and planes, we may talk of corresponding parallel axes and planes in the reciprocal lattice. This may be proved using the definition of the reciprocal lattice as a set of vectors 5,; for which the number 5, R, is an integer for any direct-lattice vector R,. But if R,,, related to R, by transformation (4.2.41), also belongs to the direct lattice, and 6, is related to b, by the same transformation, then, from condition (4.2.42), 5, -R, = 6,:R, and, consequently,

5. belongs to the reciprocal lattice, proving our assertion.

If the direct lattice possesses a symmetry plane y =z, then the plane in k space will be k, = k,. Therefore, the continuous function E(k, {) is symmetric with respect to any symmetry plane in k space. Hence on any such plane

(4.2.45)

a-VyE(k,6)=0,

with a being the unit vector of the normal to the symmetry plane. If (4.2.45) is not met, then, by symmetry conditions, the V, E(k, ¢) would undergo a discontinuity within the Brillouin zone, which contradicts the requirement that E(k, ¢) with its

derivatives be continuous in the interior of the zone. The vector V,E is normal

to the constant energy surface. By (4.2.45), it is normal also to the normal to the

symmetry

plane.

Therefore,

the

surfaces

E(k,¢)=const

should

intersect

the symmetry plane at right angles, enabling us to judge the topology of the isoenergetic surfaces (including Fermi surfaces) inside the Brillouin zone. Also, we may ascertain the general behavior of these surfaces at the boundaries of the

zones.

In this context, the concepts of contiguous and intersecting energy bands must be recalled. Figure 4.18a presents the trends in the E(k, ¢) functions along

some line in & space: a degeneracy occurs at point kg; i.e., the bands touch. Figure 4.18b depicts the case of overlapping bands. In some regions of the zone

we have E(k’,(,)> E(k",¢,), but for each particular & given the inequality E(k, €,) > E(k, ¢,) always holds. Cases are possible in which the bands both touch and cross each other.

4.2.

General Theory of the Electron Motion in a Three-Dimensional Crystal

291

ve Fig. 4.18 Energy bands in the three-dimensional crystal for two adjacent Brillouin zones ¢,

ky

and (,: touching (a) and crossing (b)

c

p

kk Oo Kk

As an example of point-symmetry applications, we examine the shape of the

constant energy surfaces when they cross the boundaries of the Brillouin zones, and also the symmetry planes inside the zones. We consider two opposing boundary surfaces of a zone, which are normal to the vectors /B and —/B and bisect them. If the point group of the crystal has a symmetry plane (I,, /,, [3), the Brillouin zone also has one; it passes through the origin and is normal to the vectors +/B. Therefore, by symmetry,

(VE-1B), = —(VE-IB),

,

(4.2.46)

where a and b (Fig. 4.19) are equivalent points on the opposite faces of the

zone. In turn, these points represent one and the same electron state; therefore

(VE), =(VE),. Then it follows from (4.2.46) that the normal derivative of the energies on both faces is zero; i.e., the constant energy surface crosses these faces

at right angles. Figures 4.19a, b show isoenergetic lines on plane k, = 0 in the Brillouin zone of a bcc lattice. All these portrayed in Figs. 4.19a, b (the inner lines, symmetry planes inside a zone with plane k, the intersections of the zone boundaries with

lines intersect the straight lines which are the intersections of the = 0, and the outer lines, which are plane k, = 0) at right angles; i.e., at

the point of intersection VE = 0. However, VE does not equal zero on all the

faces of the Brillouin zone in an fcc lattice [Ref. 4.5, Sect. 17].

a

b

Fig. 4.19. Intersection of isoenergetic surfaces with zone boundaries and symmetry planes in zones

292

4. Band Theory

4.2.5 Density of Electron States in Energy Bands.

Topological Electronic Transitions

In the band theory of solids we often have to calculate the various physical

quantities F, which are the weighted sums of the corresponding one-electron

characteristics F,(k) over the quantum states (defined by the wave vectors k) of the electron energy bands of the crystal. We did so on more than one occasion using the free-electron gas model (3.5.27, 47, 48, 81, 82). The general form of the corresponding band theory formulas turns out to be as follows:

(4.2.47)

F=2Y Fi(k) . 3

The factor 2 in this equation arises from spin degeneracy. For the sum over k to be calculated, we need to recall the definition of the electron wave vector k (Sect. 3.5). As follows from the definition, to each possible value of & in k space

there corresponds the volume 4k =(2z)°/L,L,L, = 8n°/V (with V being the volume of the usual r space). The region of volume

V, in k space contains

V,:(823/V) possible values of k and the unit volume of k space (V, = 1); i.e., the density of states in it is equal to V/8x°. Hence we may immediately write the sum in (4.2.47) over k in the following form: 2)

V F,(k) = — ~3 DFR) 4

k

ARF (k)

(4.2.48)

In the limit of infinitely large systems (V— 00 and Ak — 0) and if the function

F,(k) varies insignificantly over the interval 4k in k space, the sum in (4.2.48) may be replaced by the integral

_ 5,2 x F(k) = § dk 5 Fuk) sim The quantities F,(k) normally depend on the wave vector number through the electron energy E,(k). Pushing the analogy with the free-electron case (Chap. 3) position to determine the density g(E) of electronic levels V = 1) and in band theory. In this event the expression for

quantity F itself will be

—; F,(k)

(4.2.49) and energy-band further, we are in a (per unit volume the density of the

(4.2.50)

4.2

General Theory of the Electron Motion in a Three-Dimensional Crystal

293

or

(4.2.51)

S = J dEg(E)F(E) . Comparing (4.2.50) and (4.2.51) we obtain

g(E) = Lat)

(4.2.52)

,

where the density of electron energy levels in the ¢ band is equal to

dk g(E) = § gq OLE — F(A] . with the integral It is actually fashion. Just as in restriction that V

(4.2.53)

taken over any unit cell of k space. more convenient to determine g,(E) in a slightly different the case of a free-electron gas (3.5.5, 8), (but subject now to the = 1), we have

g.(E)dE = dN;(E) ,

(4.2.54)

where dN,(E) is the number of allowed one-particle levels, i.c., & values, energy interval between E and E + dE. As we have already seen, the number of admissible vectors & in band in the above energy interval dE is equal to the ratio of the volume portion of the unit cell in & space in which the energy lies within the

in the the ¢ of that limits

E 0, a, > 0, a; 0, (4.2.59)

describes a hyperboloid of one sheet, and for z < 0 it describes a hyperboloid of two sheets. Finally, when z =0 the topology of the isoenergetic surface also changes—its two uncoupled parts coalesce. It can be shown that the behavior of Van Hove singularities here is the same as when a new sheet appears or

disappears [4.8]. The Fermi surface plays the most important role in metals, corresponding in energy to the highest occupied level, E=E;={, at T=OK. As shown in

Chap. 3, nearly all properties of metals (except for ultrahigh-frequency proper-

ties) are determined by electrons with E = {, so that, if the topology of the Fermi surface changes, this strongly affects all the properties of the metal and, in fact, a

phase transition, called a topological electronic transition [4.9] will occur. It

may be shown that this is a 2.5-order transition, according to Ehrenfest’s classification; i.e., the singularity in the thermodynamic potential Q has the form

62 x |z|*/?9(+2z) ,

(4.2.62)

with z= E,—E,, 8(x>0)=1, and 9(x

h 2 vr

h*

+ E (q'—@"Y a? +| ry-o? |

1/2

(4.3.31)

f 1

I ! 1 1 ' t

4

|

1

y

oy

0

et

-

x

YX!

A

a

ee

t

'

'

'

1

1

1

' '

2m

ia

i

Ga

Fig. 425, Dependence of energy £ = E(8md?/h?)

on the quasimomentum & (wave vector) of the

band electron near the Brillouin zone bound-

ary. The first four zones (1, 62, £3, ¢4 and the positive values of the quasimomentum

are

Kd indicated

This dependence is portrayed schematically in Fig. 4.25. For 9 = 0, (4.3.29) goes into (4.3.25). An essential fact, not immediately seen from (4.3.24), is manifest in (4.3.31). The vector Ky and the energy E, do not vary if we displace the end of k

along the normal to the boundary of the zone; i.e., the quantity E, is determined

only by the projection of k to the boundary plane. It is only through g that the

perturbed energy depends on the normal component & to this plane. However,

the quantity y involved in (4.3.31) is quadratic [here lies the fundamental difference of (4.3.31) from (4.3.27, 28) for an unperturbed problem]. Therefore,

the normal component of the energy gradient is, by (4.3.31), equal to

GE, Oly!

fh?

av, Ey = Fa

ein h2

«(So

h?

+ oala—a'P

-ore sain et |

(4.3.32) - 1/2

}

also for y = Oanda- V,E, = 0; i.e., at the boundary (4.3.21) V, E lies in its plane, and the energy surface E = const, constructed according to (4.3.30), is normal

to (4.3.21).

308

4. Band Theory

This derivation vividly demonstrates the effect of a periodic field on the energy spectrum. Far from the planes (4.3.21), the energy surfaces do not alter in the first approximation. Before the field is turned on, the constant energy surfaces of the zone for the planes (4.3.21) touch their counterparts from the

neighboring zones. With the field on, a finite discontinuity arises between these surfaces of equal energy; they fold in such a way that they approach the plane on

which they touched earlier and have a normal tangent to that plane. This is particularly evident in the one-dimensional representation (for some one direc-

tion in & space), as is clearly indicated by the dashed segments of the curve

(parabola) in Fig. 4.25. For the values of k = an/d (n = 1,2, 3,...), where we

earlier dealt with the touching of adjacent bands (Fig. 4.18), we now have the divergence of the energy curves for adjacent zones in different directions, which leads to energy gaps. At the points of discontinuity, the energy curve approaches the straight lines parallel to the ordinate axis with the tangent parallel to the

abscissa axis.

According to (4.2.44), expression (4.3.32) defines the component of the mean electron velocity with respect to the normal to the zone boundary plane: ,. Thus, the mean velocity component goes to zero on these planes.

This comes from the “vanishing-of-current theorem” of the rigorous theory

(Sect. 4.2.3). For the one-dimensional case this is an exact relation (4.1.112). In the three-dimensional problem, the condition ¢v(k, ¢)>, = 0 on all the planes (4.3.21) is a much more severe statement than that made at the end of Sect. 4.2.3. In this drastic form, however, it is valid only in the first approximation, since,

per se, the planes (4.3.21) appear in the theory if what we take as the zero

approximation is the free electron. These planes may always be drawn, but there are no grounds for asserting that < »(&, ¢)>, on them is always equal to zero in exact theory, too. The vanishing of , stems from the fact that in the vicinity of the planes (4.3.21) the wave function in the potential field undergoes substantial changes, which can be traced with the aid of (4.3.23). Let us assume, for illustration, that E(& + g’) > E(k +g”). Therefore the sign in (4.3.30) should be positive, and we find from (4.3.23) with (4.3.24, 29, 30)

h2 B= Mee gol +[

—q'\lal

kh?

ware

iy

a? |

wy2y-1 }

a.

(4.3.33)

We see that as long as |E(k+q')-—E(k+q’")|

=

h2 7 la -—@' lal

>21Vy_gl,

either |B/a|> 1 or |a/B| > 1; ie. the linear combination

(4.3.34) ag «et Boy

is

respectively close to one of the unperturbed values ay. ¢ or Boye «. This is what

4.3.

Nearly-Free-Electron Approximation

309

corresponds to the assumption that there is no degeneracy (4.3.20, 34); Equation

(4.3.24) also yields the unperturbed energy values. In the opposite case h2

wm Ie

alll < 21%

(4.3.35)

gl ,

the amplitudes « and £, by (4.3.33), are of the same order of magnitude; i.e., the wave function turns out to be substantially different from that of a travelling wave, (4.3.33), and is a superposition of two travelling waves whose amplitudes are of the same order of magnitude. This is particularly true for 7 = 0, when (4.3.33) has the form

B=

+taVvy_g/lVe gl,

(4.3.36)

ie., |a| = |B]. Introduce the notation

(4.3.37)

Vag = [Vig lexp (ida; ) »

so that, according to (4.3.22) (with neglect of all the small corrections a’, «”, etc.),

we then have in the zero approximation

(4.3.38)

ak, BZ) = aL Sy + €XP(— iy 4)5 ygq°] » or, proceeding according to (4.2.13) to the usual expression

for the wave

function,

Walr) = afexp[i(Ko + q')-r] + exp[ — igy _g- + (Ko +q9")'r}} -

(4.3.39)

The subscript 0 of Ky here emphasizes that (4.3.39) holds for the exact equality

E(Ky + q') = E(Ky + q”). From (4.3.39) it follows that in the state described by

this function the velocity component, normal to the plane (4.3.21), is equal to zero. In addition, as an exercise, the reader can obtain from (4.3.39) a number of

derivations of the exact one-dimensional

problem

treated in Sect. 4.1.4. In

(4.3.39) the vectors Ky + g’ and K, + q” are equal in magnitude, and the vector

of their difference g’—g”

is normal to (4.3.21). Therefore the unit vectors

(Ko + 9')/|Ko + q'| and (Ky + q”)/|Ko + 9”| are interrelated as the unit vectors of an incident beam and a beam reflected from the plane (4.3.39). Thus the state described by one of the functions (4.3.39) may be considered as a reflection of the travelling wave from this plane with a phase equal to the difference of the incident and reflected waves. To some extent, this result may be viewed as the vindication of the classical “wave” treatment in Sect. 4.1.1. We should, however, bear in mind that the form of w (4.3.39) has been obtained in the approximation of the nearly-free-electron problem and this purely wave form is by no means obligatory for the exact theory.

310

4. Band Theory

We could go further and find successive approximations for y and E. But this would be pointless, for no qualitatively new result would be obtained.

It would be interesting now to consider the opposite limiting case of strong-

coupled electrons, but we wish to defer this discussion until Sect. 4.6.3. In Sect. 4.3 we noted the possibility of some compensation for the effects

which the ion cores and other band electrons have on a given band electron. As a result of this compensation, the total effective periodic potential has a small

amplitude. The fairly good agreement between the conclusions of the band model in the nearly-free-electron approximation and the experimental results

provides evidence that this actually occurs, at least in normal metals. The above

approximation may therefore be regarded not as an abstract illustration of the laws of the quantum motion of electrons in a periodic potential field but as a

good description of the real situation in crystalline conductors. In all appear-

ance, two

chief physical causes may

be pointed

out to explain

why

strong

interactions between band electrons, on the one hand, and ion cores and other

band electrons, on the other, actually reduce the potential to a weak effective

periodic potential (now often referred to as the pseudopotential, Sect. 4.6.4).

First, the Pauli principle comes into force here, preventing band electrons from being located close to ion cores, where core shell electrons are available. Second, other band electrons, owing to their high mobility, may cause the positive

potential acting on a particular band electron to decrease appreciably due to a

substantial screening effect. These two important circumstances, occurring in the ion-electron system of solids, will be discussed in greater detail in Sect. 4.6

and Chap. 5.

4.4 Effect of an Electric Field on Electronic States 4.4.1 Acceleration and Effective Electron Mass Computing the various characteristics of solids requires a knowledge of the influence which an externally applied electric and magnetic field exert on an

electron. Specifically, in order to account for the existence of metals and nonmetals and to formulate exactly the concept of the conduction electron, we must consider its acceleration in an electric field. Let us assume that at t < 0 the

electron is in the band eigenstate

Vaclrt) = ta(riexo| we now turn on a

~ perk oe | ,

(4.4.1)

constant uniform electric field F with a scalar potential

— F-r. The change of state of the electron is described by the time-dependent

Schrédinger equation

44

ih

Oy(r,t) o

where

H ery,

Effect of an Electric Field on Electronic States

(4.4.2)

(ys — CF r)(r.t)

F

has

been

defined

311

according

to (4.2.4). We

seek

(r,t)

expansion in the stationary states of the unperturbed problem (4.4.1)

W(r,t) = z Wer lty t) eg e(t) .

as an

(4.4.3)

Substituting (4.4.3) into (4.4.2), we obtain

Y be (resp| — 2h E(k’, on

za

+ O x [ Eu. C) + ih 5 Jove

- eF Wee (rexo|

= PV ape (t)L E(k’ 0’) kt

- pec

| .

(4.4.4)

Multiplying (4.4.4) by W%-(r) and performing the integration over dr, we obtain, with allowance for the orthogonality property (4.2.30),

Gay a (t) _ ie

iLeo) — E(k euch

Eeww

x (KCIPIA'C Joyee(t)

(4.4.5)

Substituting the coordinate matrix elements (4.2.35, 36) into (4.4.5) and transposing the term with 7, into the left-hand side, we find [4.15]

Oeac(t) | OFh Oaac(t) _ ot Ok

pF Ror > | plewO - B.C x f druge(r) Vaugc(r) -

bay (e

(4.4.6)

The term with {’ = ¢ may be excluded by the replacement

1

1

ieF

5 E* 0) +5 E(k, o) - oS arukc(r) Vie ugc(r) or

eFt

Oye > axexp| - >

fdrup(r) Pam

.

(4.4.7)

312

4. Band Theory

For all attainable fields F the second term involved in (4.4.7) is small and may be thrown away, for it does not lead to substantial effects. This approximation boils down to the necessity of excluding the term with (’ = ¢ in the sum over ¢' in (4.4.6). At the outset we neglect the interband transitions, replacing the righthand side of (4.4.6) by zero (it will be proved in Sect. 4.4.2 that their contribution is negligibly small). A solution to the Cauchy problem for the equation

0 eF 0 ($+ 5-5) ated =o

(4.48)

is the function Ope (t) = Oy _ ers/n,¢() -

(4.4.9)

If, with t = 0, the system was in the state (4.4.1), then, according to (4.4.3, 9),

VOD = TUDO e erin Vawnlh 0), We calculate the mean value of the velocity Allowing for (4.2.38), we find

operator

1 2E(k(t),6) ak

€0(KE, t)> = (k(t)C |B] A(e)o) = —

Fi

RUD = k + = .

(4.4.10)

in the state (4.4.10).

(4.4.11)

From (4.4.10, 11) we determine the mean acceleration

a,(kG, t) = diat)’

= y mi - '(k(t), OeF, ,

(4.4.12)

where i, j = x, y, z and we have introduced the tensor for effective inverse masses

1

mi" (ko) = 55

E(k, 0)

3k,dk,

(4.4.13)

J

For a free electron E(k) = h?k?/2m, mj' = 6,,/m [which justifies the name of

(4.4.13)]. From (4.4.12) we obtain Newton’s law ma=eF

.

(4.4.14)

Generally speaking, the tensor (4.4.13) is not a multiple of the unit tensor, and therefore the acceleration of the electron in the crystal is not necessarily

44

Effect of an Electric Field on Electronic States

313

aligned with the field. It becomes aligned when the field is oriented along one of the principal axes of the tensor (4.4.13). However, the factor of proportionality between acceleration and force [eigenvalue of the tensor (4.4.13) my '] is not necessarily positive. Near E(k, €)min all m, > 0, and near E(k, C)max all m, < 0. In the latter case the electron is accelerated not against but along the field; ice., it will behave like a positively charged particle. As noted above, these states are referred to as hole states, and band theory has thereby resolved the Hall effect “disaster” (Chap. 3). Let us assume that the electron at t = 0 has a quasimomentum hk near the

bottom of the band and is accelerated against the field. The quasimomentum of the electron varies according to (4.4.10) and, after a certain time span, emerges in the region of the top of the band; as this takes place, the sign of the acceleration

is reversed. As a result, by contrast with a free electron, an electron in a crystal, when subject to a constant electric field, performs oscillation, so that its velocity and, consequently, the current, oscillate [4.16]. This is a result of (4.2.5) and the periodicity of the E(k, ¢) function. As an example, we examine an sc lattice of period d with the field aligned along one of the translation vectors (x axis). The period t, of the oscillations in field F will then be determined by k(to) =k, + |e| Fto/h=k, + 2n/d , Le., 2nh 'o= Tel Fd

,

(4.4.15)

since the variation of k, by 21/d denotes a return to the previous state. The

typical field value for metals being F ~ 10°© V/cm, (4.4.15) yields tp + 1 s, which

is many

orders of magnitude

larger than

the mean

free time in the purest

samples. This is because the periodicity of electron motion in an electric field in metals will be completely distorted by collisions. As a result we can assume quite

accurately that the acceleration of an electron in a metal is constant, although current-density fluctuations could probably be observed in semiconductors.

4.4.2. Zener Breakdown To consider the role of the interband transitions on the right-hand side of (4.4.6),

which

were disregarded in the foregoing, we now

proceed to a somewhat

different basis for the wave-function expansion (4.4.3), namely,

vir, t)= » Bb

(heap

-53 dt'- Ey won| oO

.

(4.4.16)

314

4. Band Theory

The input equation now takes into account the variation of the electron wave

vector k->k(t) in a single-band approximation. In calculating the derivative

Ow(r, t)/dt, allowance must be made also for the time dependence of the function

Wee (r). As is clear from the foregoing, the resulting terms will cancel the contribution made by the matrix elements of the operator —eF -f, = —ieF- Vy. As a result, we obtain the following equation (4.4.6):

6 . 3p aes

eF

=

he

YJ Fe

drutiyc(r) Vetere (r) (4.4.17)

x exp] Ff de EIR’, o)— E(k(t'),compact 0

‘

where the term with {’=€ is discarded as a result of the replacement (4.3.7).

Introducing the notation

Jag (k) = — Ff drugc(r) Vane (7) and integrating (4.4.17) with respect to t, we modify this equation to

Gec(t) = HO) + YY f dtd (k(t)) PS

(4.4.18)

x exp 7) dc’ [E(k(t’), £)— E(k(t’), of Gee-(t) . 0

Neglecting interband transitions, we now substitute a, into (4.4.18) and set & (0) = 6,, (i.e. at t=O the electron is in the state |kn>). The breakdown

probability w,..; is equal to |a¢(00)|?, ie,

Wat =

J drexp iif dv’ LE(k(1’), €)— E¢&(o on 0

oO

eg

2

(4.4.19)

Further, for simplicity, we consider a one-dimensional array. We make the

substitution of the variables t +k —(eF/h)t, and replace the lower limit of the integrals (4.4.19) by — oo. We need to exclude the effects of the instantaneous switching-on of the field since, in reality, it always comes into play in an

infinitesimally slow fashion in comparison with the atomic time scale. Then nothing depends on the instant at which the field is turned on, and we may choose this instant in an indefinite past. The substitution results in h

Wao = eF

2

wo

. aki (hvexe|

i

k

J dk LER, 0)— E(k, on}|

2

(4.4.20)

44

Effect of an Electric Field on Electronic States

315

The integrand in (4.4.20) contains a rapidly oscillating term as F 0

and,

consequently, the integral can be calculated using the saddle-point method [4.17] (the first to use this for computing transition probabilities in quantum mechanics was Landau [4.18]). The fundamental contribution to the integral is

made by point ky, where the phase derivative of the oscillatory function with respect to k is equal to zero:

(4.4.21)

E(ko, 6)= Elko, n) .

In the vicinity of ky the phase varies smoothly, and far from ko it varies very

rapidly. The integrand thus reverses sign in a random fashion and is, on the average, equal to zero. Furthermore, k does not have real values (except for the degenerate case, with which we will not be concerned here) and for F +0 the

quantity w,., may be shown to tend to zero more rapidly than any power of F

[4.17]. If the point k, lies in a complex plane and the functions E(k, ¢), A,,(k) are analytic, the path of integration (real axis) may be distorted so that it passes

through ko. In this case, at ky the path of integration will have a maximum for the integrand. Therefore, up to the preexponential factors, Wye&

exp tz AaLE(KC)~ Etk nh}

2

~exp| — 2ieiF§; dx [E(k + ix, 0) — E(k + ix, mi}

(4.4.22)

The path C is depicted in Fig. 4.26; we have taken account of the fact that

eap| iF J aALEU, 0)— Elk nn} = 2

Fig. 426. Path of integration for calculating the integral (4.4.22)

and introduced the notation Ky = Imko, k= Reko. The probability of a break-

down is particularly large near binding approximation (4.3.24)

the Brillouin-zone

boundary.

E(k, 6)— E(k, n) = {(E(k+q)— E(k +q/)P? + 4[ Vy -9\?}17

=| (nae) (ea) +¢] 2\2

2

1/2

In the weak-

(4.4.23)

316

4. Band Theory

where we have chosen q = 0, q’ = 22/d, k = — n/d and introduced the notation

2|V,-4'| = 4 for the energy gap between the bands. It follows from (4.4.23) that k=

Od"

m , imdd

Qnh?’ 2

weenp| = ex

~

P|

mdd/2nh?

5

J

2

(2a?

ax| 4 -(?F*)

ape

(4.4.24)

md A?

Gel FR)

Thus the interband transition probability is, in fact, negligibly small when

md A? * Sieit

(4.4.25)

4.4.3 Quantum Theory of the Electric Inertia Effect In connection with the effective-mass concept, we may ask why the measured values of e/m in electric inertia effect experiments such as those staged by Stewart

and

Tolman

(Sect. 3.2) coincide

with

those

for

free

electrons,

and

whether or not this contradicts band theory. Such doubts, we will see, prove to be unfounded (although they are expressed in the literature from time to time [3.2]). The problem of the electric inertia effect may be treated in the most general

form, with allowance for the electron-electron interaction. Let us consider the motion of a metal, as a whole, with a varying velocity v(t). The coordinates of all ion cores will then depend on time according to the formula

R,(t) = R, + i dt'v(t') . 0

(4.4.26)

In the adiabatic approximation (Sect. 1.9) the time-dependent Schrédinger equation for the electron wave function for this nuclear arrangement has the form “

h2

Hr, N= l ~ om

_eW(r,.0) =ih

a

4;+ 2», Wr), —14) + y G(r,- ao) ve

t)

(4.4.27)

44

where W is

Effect of an Electric Field on Electronic States

317

the potential energy of the interaction of conduction electrons with

each other, and G with ion cores. We pass on to new electronic coordinates _ i dt' v(t’)

(4.4.28)

0

and represent the electron wave function as

~

i

i

Wr; = Vr, nero] 5400 +7005]

(4.4.29)

where A(t) and y(t) are, for the time being, arbitrary real-valued time functions. The transformation (4.4.29) maintains the normalization of the wave function,

and the current density of the number of particles will take the form h

i= ami % [y*(r;, OV W(r,, t)— wr; _

Vj b*(r,, t)]

ani BMH (Fete) as

*

h ~ = ami YO

w(r;, t)

OVW

~

~ ~ O— Wr OV; VG.

(4.4.30)

A(t), , alt)

toa

toe

with V; = 0/dr,. As will be seen, the transformation (4.4.29, 30) corresponds, with a specific choice of the functions A(t) and y(t), to the transition to a new

noninertial system of coordinates. In the transformation to new coordinates the Hamiltonian of (4.4.27) becomes

#'=

2mh? 4 j+

Do Wr)

+ PG(r; — RO)

.

j (r,t), remains equation. As a result, the equation for y has the form

| -3Imx Ait} Gj R)+ YW

nt gefnalie

in the Schrodinger

(4.4.36)

Besides, in view of (4.4.34, 35), 4 =f ae ale id

(4.4.37)

As follows from (4.4.34, 30), upon transformation of the wave function, j=/j' + v(t); ie., we are merely dealing with a transition to an inertial frame of reference. If v(t) = const, the Schrédinger equation for the function w (4.4.36) simply coincides with the input equation (4.4.27) for the function w. This is a

manifestation of the Galilean invariance of nonrelativistic quantum mechanics.

As is seen from a comparison of (4.4.2, 36), the acceleration leads to an “inertial force”, alternatively known as an electric inertia force. mdv Fa--—. 7 di

4. (4.4.38)

4.5

The Metal-Semiconductor Criterion

319

This expression was used in the classical treatment of the Stewart and Tolman experiments in Sect. 3.2. The formula is seen to include the free-electron

mass. However, the character of the motion of an electron subject to this fictitious field may be substantially different from the counterpart for a free electron. On the other hand, this is not essential to the interpretation of the

experiment by Stewart and Tolman, since they utilize an experimental value for the conductivity that is certainly determined by the effective mass rather than the free-electron mass.

4.5 The Metal-Semiconductor Criterion 4.5.1

The Metal-Nonmetal Criterion in Band Theory

So far in this chapter we have been considering the properties of one-electron

states. However, describing the properties of solids even in terms of the band model necessitates allowance for many-particle effects. In the simplest approxi-

mation one completely neglects the dynamic electron-electron interaction and takes into account only the statistical correlation, which is due to Pauli’s exclusion principle and is described by the Fermi-Dirac function (3.5.9) (the scope of applicability of this approach is discussed below; this is actually one of the central problems of the entire theory of condensed media). The heat capacity, paramagnetism, etc. are determined in the same way as in electron gas theory (Chap. 3), but with the “band” density of states (4.2.58). Of supreme importance to the metal-nonmetal criterion are the electrical properties (Chap. 3). In the ground state (J= 0 K) metals possess free electrons, whereas no free electrons are available in nonmetals. Free electrons are accelerated by a weak electric field. In such a field we may neglect the interband transitions, the probability of which, according to (4.4.25), is exponentially small, and replace k(t) by & in (4.4.12). If the electrons do not interact, the result for the acceleration of all of them (equal to the time derivative of the total current) will be

W, = Faulk )n(ke) = k

Jj

eF;¥ae my *(ko)n(ke) ,

(4.5.1)

with the summation over k being carried out over the first Brillouin zone. For

T=0 K

the n(k€) satisfies the plot given in Fig. 3.6. If, for some C, the quantity

E(k, ¢) is less than or equal to E, for all k, this band does not contribute to (4.5.1), since

x mj" (ko) =0 .

(4.5.2)

320

4. Band Theory

This can be proved by substituting (4.2.20) into (4.4.13); as we do so, the term

with R,,=0 cancels out, and for R,, #0 we have ¥ exp(ik-R,,) =0 .

(4.5.3)

.

Formula (4.5.3) may be proved as follows:

¥. exp(ik- Ry) = —Ra?Y, Apexp(ik- Ry) E

E

dk —_R-2 =—R,7V J (on) A,exp(ik-:ip. R,,) dS,

.

= —R,;? Vos oes V,exp(ik: R,) d

= —iR,,R,? Vol eonexptik- Ry) =0 . Here BZ denotes the first Brillouin zone, { dS, is taken over its boundary, and

dS, is an element of area in space, multiplied by the unit normal. In the transformation from the volume integral to the surface integral we have exploited Gauss’ theorem; the latter integral is equal to zero, since at the opposite

zone edges 1, 2

dS, = — dS,,; exp(ikz" Ry) = expLi(k, + bf): Ry] = explik, Ry) Thus the electrons of completely filled or empty bands are not accelerated by an electric field. This is readily interpretable. With neglect of the zener breakdown, the total current of {-band electrons is equal to

aed, o(& + oe t)n(kE) =e v(k, ont - oe ‘) k

k

(4.5.4)

[see (4.4.11)]. An electric field causes quasi-rotation of the distribution function in k space—k +k —eFt/h followed by reduction to the first zone. If n(k¢) = 1, J, will be independent of t and at time zero is equal to zero by symmetry considerations: E(k, ¢) = E(—&, ¢); whence v(k, ()= — o(—k, ¢); consequently, i(t) =0. ‘ Thus, in band theory, a solid is a metal only if some of the energy bands are

partially occupied or overlap. For example, any crystal that contains an odd number of electrons per unit cell may be a metal.

As it is, the total number of states in a band is even, since a twofold spin degeneracy occurs [for electrons that do not interact dynamically, their energy is

4.5

The Metal-Semiconductor Criterion

independent of spin (4.2.4)]. For a metal though, there is electrons, and therefore one or several bands will be partially if we assume that the 3s states of Na form an energy band problem posed in Sect. 4.1.3), the latter is exactly half full.

321

an odd number of filled. For example, (in the spirit of the On the other hand,

solid hydrogen (at least at pressures that are not too high) is not a metal. There is one electron per hydrogen atom, but it is energetically more favorable for the

atoms to combine in H, molecules. As the hydrogen atoms do this, they occupy sites that are obviously nonequivalent (the interatomic spacing in much smaller than the intermolecular spacing), and the number of unit cell turns out to be even. By contrast, the number of electrons even; nevertheless, solid Ca is a metal. Thus, some energy bands

a molecule is electrons per in, say, Ca is overlap and

therefore are only partly filled. Figures 4.27a—d provide a schematic picture of the energy bands and the filling of the energy insulator, a semiconductor, and a metal.

levels with electrons for an

Fig. 4.27. Energy band filling (shadowed) in the ground state of an insulator (a), a semiconductor (b), and a metal (c, d)

The above metal-nonmetal criterion considered by Wilson [4.19] in band theory accounts for a large body of experimental evidence and is the basis for

considering nearly all of the properties of solids. However, a number of prob-

lems arise with the generalization of this criterion for noncrystalline condensed states (liquid mercury is a metal, whereas water is not) and with allowance for the electron-electron interaction. For example, experiment provides conclusive evidence for the fact that in most rare-earth metals 4f electrons do not

participate in conductivity, although the 4/ band is only partly filled. This is due to their strong Coulomb repulsion and the very narrow 4f band. Although formally these problems lie outside the scope of band theory, they will be allotted some space here in view of their supreme importance.

322

4. Band Theory

Nonmetals

are

divided

into

insulators

(dielectrics)

and

semiconductors

(although this has no profound physical meaning and is only based on con-

vention; see Sect. 1.8). Solids in which, in the ground state, the energy gap G between the last occupied (valence) band and the first empty (conduction) band is not larger than 1 to 2 eV are called semiconductors. Solids with G 2 2 eV are

said to be insulators. Special mention

must

be made of gapless (zero-gap)

semiconductors (gray tin, HgTe, HgSe, etc.), where the conduction band and the valence band touch at one point, determined by the symmetry properties of the crystal (Fig. 4.28). When in the ground state, these materials are nonmetals, since

they have no partially filled bands. However, near the ground state the energy spectrum of these zero-gap semiconductors is continuous, just as in metals.

Fig. 4.28 Energy spectrum of a gapless semiconductor

This formulation of the criterion for differentiating between the state of a

semiconductor and that of an insulator is only a convention. A more distinct and physically clear formulation to distinguish between insulators and semi-

conductors

is to consider their electrical conductivity.

Insulators are then

viewed as crystals that exhibit chiefly ionic conductivity (a small admixture of electronic conductivity at very high temperatures is immaterial). Semiconduc-

tors are regarded as crystals that exhibit, long before the onset of observable

ionic conductivity at very high temperatures, a pronounced electronic conductivity whose temperature dependence is directly opposite (in sign of the temperature derivative) to that of metals. This difference in the physical nature of the

electrical conductivities of an insulator and a semiconductor arises from the difference in the size of the energy gap between the occupied valence band and

the empty conduction band at T= 0K.

We now evaluate the temperature dependence of the number of current carriers in pure (intrinsic) semiconductors. To do this, we determine their

chemical potential ¢. The energies will be counted from the top of the valence band. The number of thermally excited band electrons at temperature T is

equal to

De

—_f

Ne= f0aEaAE)| exp ES) kyT

+ |

-1

,

(4.5.5)

where D, is the bandwidth, and g,(E) the density of states in it. Assuming that

G-C>kgT,

(4.5.6)

4.5

The Metal-Semiconductor Criterion

323

we find

N, © exp ( (-G kaT i 0 dEg,(E)exp (—E/kpT) .

(4.5.7)

Similarly, the number of the unoccupied states (holes) that have arisen in the

band is equal to N, ® exp (—

Dn

C/kgT) J dEg,(E)exp(— E/kgT)

,

(4.5.8)

where D, is the width, g,(E) is the valence-band density of states, and it is assumed that C>kyT.

(4.5.9)

Evidently, in an intrinsic semiconductor

N.=N,.

(4.5.10)

As has already been noted for phonons in Sect. 2.3 (see Sect. 3.5.8), we may assert that near the band edge in the three-dimensional case

g.(E) = A,E'? ,

9,(E) = A,E"? ,

(4.5.11)

with A, and A, being some constant quantities. For kg T < D,, D, a small E makes a fundamental contribution to the integrals (4.5.7, 8). Substituting (4.5.11) into (4.5.7, 8), we find from (4.5.10)

G kT, A, cxst Sint.

(4.5.12)

€

Conditions (4.5.6, 9) evidently hold for { = 1 eV and G = 2 eV and at a reasonable temperature. Now we substitute (4.5.11) into (4.5.7): De

N, = Ny = exp(—G/2kgT)(A,A,)"? [ dEE'exp(— E/kpT) oO

(4.5.13)

= exp(—G/2kgT )(AyA,)'7*(ky T )?? DelkpT

0

dxx'2e-* ~ (nA, A,/4)"/2 (ky T)?/2exp(—G/2kgT)

(0. DkeT,

f dxx'?e-* = vin) ; 0

,

324

4. Band Theory

Formula (4.5.13) gives the temperature dependence of the number of current

carriers for intrinsic semiconductors. The electrical properties of real semi-

conductors, however, are determined chiefly by impurities. If we implant, say, an As atom into a Ge crystal, the following will occur:

The Ge atom is quadrivalent, so each site in its crystal (diamond-type lattice,

Fig. 1.22) has four nearest neighbors. The As atom has five valence electrons, which are bound to the nucleus comparatively weakly. If the Ge atom in the

lattice is replaced by the As atom, four electrons serve to form valence bonds

(Fig. 1.34), and the fifth may become practically free. This is so because of the following: The radius of the bound state of an electron with positive charge |e| in vacuum is on the order of h?/me?, and the binding energy is known to be equal

to AE = me*/2h? = 13.5 eV. In a medium the interaction of charges becomes ¢

times weaker: e? — e?/e, where ¢ is the static dielectric constant, and the mass m

should be replaced by the effective mass m*. The radius of the bound state is then 2

2

axle =m F305 m*e? — m* me?”

A,

(4.5.14)

and the binding energy is m*

me*

4E =~ a pz = 13.5 meV.

(4.5.15)

For semiconductors the constant « is rather large: ¢ 2 10. This is due to the small band gap of G; when G = 0 the system becomes a metal and & = oo. As will be seen in Sect. 4.6.6, the smallness of G leads also to a small m*/m. As a result, a may reach a magnitude on the order of dozens of interatomic distances (this is what allows us to describe the influence of the medium of the macroscopic, i.e., averaged, quantity c). The binding energy, for example, for an As impurity in Ge, diminishes in comparison with the ionization potential of As in vacuum from 9.8 eV to 0.013 eV. Such “shallow” bound states are easily destroyed by thermal

motion already at T ~ 100 K. Impurities that donate electrons are said to be donors. We may consider in an analogous fashion some trivalent impurity in Ge that supplies holes to the valence band (Fig. 1.30). Such impurities are called acceptors. Figure 4.29 is a sketch of the energy spectrum for an extrinsic semiconductor.

The donor levels and all electron-occupied levels at T= 0K lie beneath the

conduction band and are separated from it by a small distance (their ionization energy ~ AE), whereas the acceptor levels, not occupied by electrons at T = 0K, overlie the top of the valence band and are separated from it by a small distance (their ionization energy E, counted from the top of the valence band, E,). Later we will return to some of the fundamental points of the theory of semiconductors. More detailed information on this area of solid-state theory is available in the literature [2.34; 3.19; 4.20, 21].

4.5

The Metal-Semiconductor Criterion

325

4Ed

G oa

oO

0

oO

a

\\

45.2

aEa

Fig. 4.29. Energy

spectrum

of a doped

semi-

conductor with specification of acceptor AE, and donor 4E, levels

The Peierls Transition

Up to this point we have assumed that the lattice is rigid and the electrons do

not interact. The fundamental problem of the metal—-nonmetal criterion is that these two assumptions underlying the band model are liable to incur gross

errors. Here

we concentrate

on a purely qualitative treatment

of the most

important points. To start with, we relinquish the assumption of the lattice being

a

rigid (as before, the electrons are assumed to be noninteracting). We examine a linear array with one half-full energy band (Fig. 4.30a). In the one-dimensional case there are (Nd/2z) Ak states (with d being the lattice period) for the interval of values of the quasiwave vector 4k (Chap. 2). Therefore, the occupied states in the half-filled case are those with k < n/2d (Fig. 4.30a). We consider a minor

Fig. 4.30. Lowering of the total energy of band 41,

K.

electrons

when

the

lattice

period

is

doubled

(Peierls transition): prior to doubling (a), subsequent to doubling (b)

326

4. Band Theory

distortion of the array, where every other atom shifts slightly from its equilibrium position. The lattice period then is not d any longer, but 2d (Fig. 4.30b), and

a discontinuity occurs at the new Brillouin zone boundaries (Sects. 4.1, 3). As this takes place, the energy of all the occupied states lowers somewhat, and that of the unoccupied states rises (Fig. 4.30b), leading to a gain in the total band

electron energy in the array and, consequently, to a decrease in the total energy

of the one-dimensional crystal [Ref. 2.7, Chap. 5]. Broadly speaking, this distortion may also entail an increase in core electron energy. However, in

metals this is small compared with the metallic bond energy due to conduction

electrons. Thus distortion has led to the occurrence of an energy gap at the Fermi level, and a metal has turned into a nonmetal. Clearly, another distortion will not lead to an appreciable energy gain, since the energies of the states above the gap increase by approximately the same amount as that by which the energies of the

states below the gap decrease (4.3.24). Obviously, for any band occupancy it is possible in the one-dimensional case to choose a distortion such that a gap opens at k = k,. The corresponding period of the resulting “superlattice” will be

D = 2n/dke .

(4.5.16)

If D is commensurate with d, i.e. D = pd/q, where p and q are integers that have no common divisor (p > q), the resulting “superlattice” has a period pd. Here

Bloch’s theorem is applicable and we can exploit the theory outlined in this

chapter.

But

if D and

d are incommensurate,

we

are confronted

with

the

complicated and little-explored problem of the electron motion in a nearly

periodic potential [4.22]. The previous treatment (although not totally rigorous) shows that in the ground state a one-dimensional array cannot be a metal. This conclusion is supported experimentally for some quasi—one-dimensional systems (i.e., those consisting of arrays with a weak interaction between them), which are semiconductors [with a superstructure (4.5.16)] at low temperatures and change into metals as the temperature is raised (Peierls transition).

In the three-dimensional case, generally speaking, it is impossible by distor-

ting the lattice to attain a substantial lowering of the energy of all the occupied

states (this requires that the Fermi

Brillouin-zone with the zone decreasing the situation, only may occur and

surface coincide exactly with the new

boundary). Nonetheless, if the Fermi surface nearly coincides boundary, the total energy gain attained in the crystal by band energy at the points of contact may be appreciable. In this a local spectrum readjustment at determinate points of k space so there is no metal-nonmetal transition.

These qualitative considerations enable us to understand

the empirical

Hume-Rothery rule [1.32, 3.16] relating to the structure of some alloys that arrange themselves in an ordered fashion (for example, Cu-Zn). A typical feature

of such alloys is that at certain concentrations of the constituents, superstructure phases

are

produced

in which

the

ions of different

elements

are regularly

4.5

The Metal-Semiconductor Criterion

327

arranged. According to this rule, the stability limits of the various phases (bec, fcc, etc.) correspond to quite definite mean electron concentrations that are very

close to the values for which the Fermi sphere touches the bands for the corresponding structures on the inside of the boundary surfaces. [The spherical

Fermi

surface

approximation,

which

corresponds

to

quasi-free

electrons

(Sect. 4.3), gives a fairly good fit to experimental data for many real metals.] The energy gain due to the local gap arising in the electron spectrum at the points of

contact stabilizes the relevant phases. The considerations set forth above show the importance of the contribution which noncentral forces can make to the lattice energy of the metal (with a fixed lattice structure the band energy of electrons certainly cannot be represented as

a sum of pairwise ion-ion interaction energies). For reasons close to those outlined above, some metallic-bonded compounds undergo structural tran-

sitions accompanied by distortion of a highly symmetric initial lattice. The

character of these distortions is associated with the shape of the electron energy

spectrum.

4.5.3 The Mott Transition Here we will try to ascertain how seriously the neglect of the electron-electron

interaction in band theory may affect the metal—-nonmetal criterion. Part of the

electron-electron interaction may be taken into account by including it into the crystalline potential (self-consistent field, Sect. 4.6.1). The many-particle effects that cannot be included in this way are said to be correlational; we concentrate

on these effects in this section. We examine a crystal with the number of electrons equal to the number of

sites. According to band theory, the ground state of this many-electron system is metallic and has a half-full energy band, there being two electrons with antiparallel spins in each orbital (Bloch) state (for simplicity, we exclude the bandoverlap case). The probability of finding on some crystal site an electron with

spin up is equal to 1/2, that for an electron with spin down is the same, and, consequently, the probability of finding on some site two electrons with opposite spins is equal to 1/4. This state is undoubtedly advantageous in terms of a gain in band (kinetic) energy, since the lowest one-electron states are the ones occupied. However, the large share of doubly occupied sites leads to an increase

in electron-electron repulsion energy in comparison with the so-called homo-

polar state electron in Heisenberg an increase

in which one electron sits on each site. On the other hand, the this state is localized in a small volume, which, according to the uncertainty relation, leads to large fluctuation of momentum and to in kinetic energy.

We label the characteristic Coulomb repulsive interaction energy by U, and

the bandwidth

(i.e., the characteristic kinetic energy of electrons) by

W. The

328

4. Band Theory

above lines of argument show that with U < Wthe interaction will lead only to some small corrections to band theory, but, on the other hand, when U 2 W, the ground state changes radically and each electron is localized on its site. Clearly,

this state will be nonconducting. (Incidentally, a rigorous proof does not exist as

yet. Moreover, the metallic state in the one-dimensional case turns out to be unstable for an arbitrary U [4.23].) Somewhere at U = W a metal-nonmetal

transition should occur [4.24], either smoothly or sharply. All this remains to be

confirmed experimentally (4.25, 4.26]. Thus the electron motion in narrow energy bands differs drastically from

that predicted by band theory. Realistically, such narrow, partially filled bands

only originate from d or f states. Classical examples of materials that should be metals in keeping with band theory but actually are not, because of the correlation, are some transition-metal oxides such as NiO (for more details see Mott [4.27]). Such materials are said to be Mott insulators, and the metal-

semiconductor transition due to correlation is called the Mott transition. Thus

far, the theory of this transition has not been constructed in a form that is in any way complete; some of the related problems are discussed below.

45.4

Disordered Systems

Here we consider the result of abandoning another fundamental assumption of band theory, namely, strict spatial periodicity and, as a consequence, Bloch’s

theorem. Experiment shows that, in their electronic properties, liquid metals

differ little from crystalline metals. This phenomenon was qualitatively explained by Shubin [1.41]. According to Shubin, for the usual concepts of the quantum theory of solids to be applicable to systems devoid of spatial periodicity, two conditions must be fulfilled: First, there should exist steady currentcarrying states. Second, the potential fluctuation scattering in the system should not be very strong (in today’s more refined form this condition says that the mean free path length should be large compared to the electron wavelength) so

that the lifetime of an electron in the eigenstate is much larger than the collision time. The first requirement holds in the three-dimensional case if the potential itself is sufficiently small. To prove this, it suffices to apply perturbation theory

for a continuous spectrum. The wave function in a weak potential has the shape of a plane wave with small additions and, consequently, the can carry current. Let us emphasize, however, that in dimensional cases this correction, found from perturbation a radical readjustment of states may be expected even for

corresponding state the one- and twotheory, diverges and a weak potential.

The problem as to why the potential in liquid metals (and in metals in

general) may

be regarded as small was touched

on in Sect. 4.3 and

will be

discussed in some detail in Sect. 4.6.4. The smallness of the potential also assures the fulfillment of the second condition.

4.5

The Metal-Semiconductor Criterion

329

Now we proceed to a case which violates the second condition; i.e., the free

mean path length is comparable with the electron wavelength (the latter in

metals corresponds to the interatomic spacing). The wave functions (x) of electrons on neighboring sites will be totally uncorrelated (Fig. 4.31). Anderson

[4.28] has shown that with a large amount of disorder, the electrons turn out to

be localized in some region of space and their states, therefore, do not carry

current.

Fig. 4.31. Shape of the wave function y(x) with

strong disorder

Consider

a liquid

or amorphous

semiconductor.

In a crystalline

semi-

conductor the density of states has the shape portrayed in Fig. 4.32a, showing

the presence of an energy gap. In a disordered system the energy of an electron on a site fluctuates as a function of environment, which results in the band edges

being smeared. A so-called pseudogap forms (Fig. 4.32b). The states at the center of the pseudogap are formed by a small number of sites, because these states require large and therefore improbable energy fluctuations. The distance be-

tween the centers is on the average large and the wave functions of the electrons sitting on them do not overlap. Therefore, the states with the energy lying at the center of the pseudogap are localized, whereas the states at the edges of the

pseudogap are, in some way, more similar to the usual band states. Critical energy values exist separating localized states from current-carrying states (such values are called mobility thresholds). If, due to a change in electron concentration or under the effect of high pressure, the Fermi level passes through the mobility threshold, the insulator (semiconductor) will turn into a metal or vice versa. This situation is called an Anderson transition (Sect. 4.9).

ge)

LL

E

E

Fig. 4.32. Formation of a “pseudogap”: density of states in energy positions as a function of energy, in the presence of a gap (a), in the presence of a pseudogap (b)

330

4. Band Theory

This concludes our discussion of “nonband” effects in connection with the fundamental problem of the metal-insulator criterion. Now that we have

reviewed the range of applicability of band theory we proceed to a consideration

of some of its specific problems. Although modern solid-state theory goes far

beyond the scope of the band model, the latter has a sufficiently broad spectrum

of applications, primarily for normal metals and crystalline semiconductors.

4.6 Computing the Electron Energy Spectrum of Crystals 4.6.1 Self-Consistent Field Approximation As stated previously, the interaction between electrons in solids may not, generally speaking, be viewed as weak (at least it is not weak in atoms that constitute a solid). Therefore, any energy spectrum calculation that has pretensions at least to be qualitatively in agreement with experiment has to allow for the electron-electron interaction in a real solid. On the other hand, because of the complexity of the many-particle problem, some results can only be obtained if one formally preserves the scheme of the band model, i.e., by setting up a oneelectron, wave-function Schrédinger equation, which only partly takes into account the effects of Coulomb repulsion [4.29]. We consider the many-electron Schrédinger equation for a crystal (1.9.6). In this equation we first make the replacements G(R;, r;)— V,(r,) and Wr, ry) > e?/|r,—ryl. Intuitively, the possibility of treating the nuclei as

“frozen” seems obvious (Sect. 1.9). However, this (adiabatic) approximation is very difficult to substantiate rigorously. (The problem is considered in greater detail in Chap. 5.) Equation (1.9.6) is equivalent to the extremum condition for the functional ¢ ¥| |?» for a particular wave-function normalization given 6

x

fdr, ... dry P*(r,0,,.... yon)

ett? (6

YX

— E) P(ryo,,-. 6, tytn)

fdr...

=,

(4.6.1)

dry|P(rio,,..., even)? = 1,

where 6 is the variation symbol. By varying « ¥| | ¥ > — EC ¥|¥) with respect to ¥*, we obtain a Schrodinger equation. The E plays the role of an indefinite Lagrange factor. According to the Pauli exclusion principle, the function ¥ is antisymmetric with respect to permutations of the coordinates and spins of any two electrons

4.6

Pdi.

ie

ye

Computing the Electron Energy Spectrum of Crystals

Gn) =

— PUG.

6 Oj -+ U- + An) »

331

(4.6.2)

where the notation introduced is q; = (r,o;). For noninteracting particles P (41, 92, ---»4w) has the form of the determinant involved in the one-electron functions ¥,(q): Wilds)...

Pd, --- dw) = (NI?

W, (an)

W.(41)--- Wys(4n)

Wov(41)+» - WonlGn)

= (N71? YF 65 P Wi). - Won(Gw) +

(4.6.3)

with v, being the quantum numbers of occupied electronic states, and P the

permutation of the indices v,,..., vy, the quantity ¢; parity of permutation. We make use of the direct variational method; i.e., the functional under the variation sign in (4.6.1) on a form (4.6.3), where w,(q) are some trial functions. quasi—one-electron Hartree-Fock (or self-consistent

= + 1 depending on the

we seek the extremum of class of functions of the This will give the best field) approximation in

terms of the variational principle. First we calculate the normalization integral

CHIP) = Jdqy... dayl¥(4i -- - aw)I? =(NI7! x egtsfdqy... dan(F'v3,(qi)-

(4.6.4)

WEG) LFW, G1) --- Wew(Gn)]» where j dq, = )\,, J dr,. We now require that the additional conditions

(4.6.5)

J dqv3(qv.(q) = 5 be fulfilled. Only the terms with

¥ = ¥’ then contribute to (4.6.3), all of them

being equal to unity. The number of the various permutations F is equal to N!, so (4.6.4) is automatically equal to unity. We now calculate the functional < ¥|2/|¥ >. Consider, as an example, the most complicated term

=(N)'

YS ges

at

“Sdqy... dqylF'W8. (41)... WE aw) PW, (ai) - - : Yon(Qn)]3 2 Ies—

vt

(4.6.6)

332

4. Band Theory

The terms with q;, q; in (4.6.6) will be separated out and integrated over the other variables with allowance for (4.6.5). The contribution to (4.6.6) is made by two sets of permutations 9’= ¥ and 9’= 9, where # differs from # in that it has the permutation v; = v,. Here e3 = 1, egeg = — 1, and we have

CHIE Inn) = 4 J dada'ysia) Wear — 61 Wa) Again, the factor Analogously,

(N!)~'

(46.7

— vg)

cancels

out

by

combinatorial

considerations.

CYIEIP) = Vf dqut(q) Hova) 2

(4.6.8)

+ sz dq dqpt(qv2(q)\r—r\"* “LH b-(9) — va

,-(9)1 .

2

Hy=- ~ A+V,(r). Taking into account (4.6.5), we solve the variational problem

ar@| PI¢

—

YdYA, [|J dqwt avtiav(a,

4.6.9 (4.6.9)

=0.

With allowance for (4.6.8), and carrying out the variation, we obtain

[- Fat ninse sar 20 ofr) 21 | bate ) YS fdr bh (ro) (r'o')|r—r |" —2 ive’ “Wis(ro) = y Ais, js Vis (ro) Jjs

.

Here is = v, js’ = v’, where i, j are orbital quantum quantum numbers; and

er) = Yb, (ro)|? jso

(4.6.10)

numbers, and s, s’ spin

(4.6.11)

4.6

Computing the Electron Energy Spectrum of Crystals

333

is the total electron density at point r. At the moment we are taking into account

the possible dependence of the orbital part of the wave function on spin projection (spin-polarized self-consistent field, or spin-unrestricted Hartree— Fock approximation). Perform now the unitary transformation of the functions

Wis(ra):

¥9) = Le V.@ »

vil = Lew

-

(4.6.12)

Here

HCL ACE

CACO

(4.6.13)

Using the unitary properties of the matrix ||c,, ||, we have

Y chevy = Ow >

(4.6.14)

“

which allows us to obtain (4.6.12, 13). Substituting (4.6.12, 13) into (4.6.10), we arrive at the result that (4.6.10) maintains its form if we make the replacement

UW

Av

Diy Ch AwCwy. By virtue of the hermiticity of | 4,,-||, evident

from (4.6.10), a unitary matrix ||c,,|| exists that diagonalizes |{/,,||. Only this representation will be used. Denoting the eigenvalues of the matrix ||A,,|| in terms of ¢, and omitting the tilde, we find h?

dr'

g(r

|- 5 A+ Vale) + ef re Vue -é >» fdr WE(r'o'\Wil’o’)Ir—r'|- jer) = Wilt).

(4.6.15)

Equations (4.6.15) are said to be Hartree-Fock equations. The third term

enclosed in square brackets is a Coulomb potential, which is produced by all the other electrons acting on a given electron [4.30]. The last term on the left-hand side of (4.6.15), called the exchange interaction, is due to the Pauli exclusion

principle and is of a purely quantum origin [4.31]. This interaction is nonlocal in character. Therefore, the relevant equations are very complicated, and currently are not often used to compute the energy spectrum of solids. Before passing over to further simplifications, we wish to discuss the physical meaning of the parameters ¢,,. Multiplying (4.6.15) by w%(rc), taking an integral over r, and carrying out summation over o, we find, with allowance for (4.6.5),

bi. = [islis] + >: { Lis, js’lis, js’] — Lis, js'|js’, is]}

#

(4.6.16)

334

4. Band Theory

where the notation introduced is

[a1B) = f dqu3(q)%o¥,(4) »

[aBly5] = ef dadq'wS(awF(air—r'l'¥,(a)valq’) -

(4.6.17)

The quantity (4.6.17) is equal to the contribution that the terms depending on

the coordinates of an electron in the state |is) make

to the total energy

. Therefore, this quantity is equal, with the sign reversed, to the energy required to remove an electron from the state |is>, if we neglect any resulting variation of the wave functions of the other electrons (Koopmans’ theorem).

Let us discuss the meaning of the restriction made. The removal of one electron leads to a relative variation of the wave functions of all the other electrons by an amount oc N ~!. Since the first variation of the total energy with respect to the wave functions is equal to zero, the change is energy due to the variations of w,, by an amount

o«N~!

constitutes oc N~?. When

N functions

are varied, this change in energy is on the order of N~'; i.e., it is negligibly small [4.32].

We consider N states and N electrons, and for this reason all the states are occupied. Correlation effects may lead, however, to partial filling of some states (as with d and f metals and many of their compounds). Here the entire scheme of

our treatment becomes dubious. For example, the 4f electrons in a Eu atom ina

EuO crystal, hardly sense the presence of electrons on other atoms (because of the small falloff radius of the wave function), while interacting very strongly with other 4f electrons in the same atom. Therefore, the variation of the wave

function of the other “essential” electrons when this electron is removed is large. The conditions under which Koopmans’ theorem and the Hartree-Fock method in general may be applied to compounds of d and f elements are not

clear. On

the other hand, computational

practice shows that some energy

characteristics of these materials, calculated using some version of the selfconsistent: field approximation, give a fairly good fit to experiment [4.33, 34].

As mentioned previously, the Hartree-Fock equations are complicated for band calculations. A certain inconvenience in purely theoretical terms is caused

by the fact that the effective Hamiltonian itself depends on the number of the state for which the calculation is performed. Therefore, the last term on the lefthand side of (4.6.15) is replaced by some potential that depends only on the density of particles with a certain spin projection

2

{- us A+ V,(r) +e? fdr’ 2m

+

e(r’) Ir—r'|

LO; (r), Qe, cor} vate

= €sWi,(ro)

(4.6.18) .

46

Computing the Electron Energy Spectrum of Crystals

335

The quantity v,, is called the exchange-correlation potential, and the method

based on the replacement of the nonlocal exchange term in (4.6.15) by some averaged potential is referred to as the Hartree-Fock-Slater method.

The nature of the exchange-correlation interaction may be understood from

the requirement that the wave function be antisymmetric (4.6.2), from which it follows that for electrons with parallel spins the probability of their being located at the same point is equal to zero. Therefore, by including the potential

produced by spin-up electrons and the potential produced by spin-down elec-

trons into the self-consistent field acting on a spin-up electron, we reassess the

Coulomb repulsive interaction energy of electrons with parallel spins, since, on the average, these electrons are separated by large distances. The radius of what is known as the Fermi hole around an electron with the spin oriented upward—

i.e., the radius of the region in which there are no other such electrons— is on the

order of [e,(r)]~'/, the corresponding gain in Coulomb energy being about

e?[e,(r)]'®. The so-called Slater Xa method employs the equation

[-

h?

xn 4+

g(r’) V(r) + e?§ drPF — 3ae?

3

(72) which

1/3

(4.6.19)

Joan = epi lr) »

is similar for a spin directed downward.

numerical parameter, which is somewhat

The a in this equation

is a

less than unity. For the rules of

selecting « values, other problems of the Xa method, and its relation to the Hartree-Fock method, we refer the reader to Slater [4.33]. Upon solving (4.6.19) for the initial choice of @,, @,, the functions ,, thus found are used to construct new Q;, Q,, etc., until a self-consistency of the required accuracy is attained. In a self-consistent solution for d and f elements we sometimes have ¢,, # €;,, Q; # Q,. This indicates the existence of magnetic moments and some magnetic ordering, which are not considered here. We confine ourselves to a system with @; = @, = Q/2; the spin subscripts of y, and ¢; will be omitted. In this case (said to be spin restricted)

YLE(r)] = — 320 ( 200)

1/3

.

(4.6.20)

A major drawback to the Xa method is that the correlation of electrons with antiparallel spins is neglected. A great deal more complicated and exact ex-

pressions have been proposed for the exchange-correlation potential; we do not consider them, for they require the use of many-body theory [4.35-37].

336

4. Band Theory

4.6.2 Solving the Schrodinger Equation. Formulation of the Problem and the Cellular Method We now investigate practical methods of solving the Schrédinger equation (4.2.4) for a particular given potential. According to Bloch’s theorem (4.2.10), for the form of the wave function y,,(r) to be known everywhere, it suffices to know

its form in any one of the unit cells of the crystal. This function is chosen in its most symmetric form—as a Wigner-Seitz cell—which in the direct lattice is constructed in the same way as the first Brillouin zone in the reciprocal lattice (Sects. 1.2, 4.2.2). The boundaries of this cell r, are given, similar to (4.2.24), by

rR, = tHIRI ,

(4.6.21)

where R; is the translation vector connecting a Bravais lattice site with all the neighboring sites. The wave function satisfies Bloch’s theorem, so the following boundary conditions are imposed on the values of its first derivative at the boundary of the cell,

Waclr + R) =e y(n) , OWec(r + R) = On(r + R)

gitR OW g(r) On(r) ’

(4.6.22)

where r and r + R belong to opposite cell faces, R is the translation vector that connects them (Fig. 4.33), and 0/dn is the derivative with respect to the normal to the cell surface, with a(r + R) = —n(r). The Schrédinger equation is of second order, and therefore the boundary conditions for Wg, and dy,4c/n suffice; for higher-order derivatives they will be fulfilled automatically. °

o °

°

°

INL. R

°

°

Fig. 4.33. Derivation of the boundary conditions on the surfaces of the Wigner-Seitz cell

The transition from the Schrédinger equation in the crystal to the equation in the cell is exact. When band theory was in its infancy, an important tool was the Wigner-Seitz approximation, which enable estimation of the binding energy of alkali metals. In the Wigner-Seitz approximation the cell is replaced by a sphere of equal volume (of radius r,). It is postulated that the potential in this

4.6

Computing the Electron Energy Spectrum of Crystals

337

sphere is spherically symmetric, and the case of k = 0 (bottom of conduction band) is considered. Then (4.6.22) holds if y depends only on |r| and satisfies the condition

aviirl) FT

lrl=r, .

(4.6.23)

This problem differs from the definition of the energy of the s state of the atom only in that it has the boundary condition (4.6.23) instead of the condition W(Ir|) +0 for |r| oo. This change results in a shift of eigenenergies. By lowering the energy of, say, the 3s level of Na (bottom of the 3s band) in

comparison with the energy of a free atom, we can estimate the binding energy per atom,

and, by varying r,—i.e., actually the density—and

exploring

the

energy variation, we can evaluate the compressibility, in fairly good agreement

with experiment. The Wigner-Seitz method, however, has a very narrow scope

of applicability and, with the advent of powerful computers, has been superseded by more exact methods, although they are more cumbersome computationally.

If we expand the required function in some orthonormalized set of functions satisfying (4.6.22), the Schrédinger equation will transform into an infinite

system of linear algebraic equations for the expansion coefficients. The ordinary approximation is to “truncate” this expansion. We wish to enlarge upon this problem.

As discussed in Sect. 4.6.1, the Schrodinger equation is equivalent to the variational principle

5 f dry*(r)( —E)W(n) =0, RQ

(4.6.24)

where # isa one-particle Hamiltonian (4.2.4), and the integration is taken over the Wigner-Seitz cell Q. As in Sect. 4.6.1, we assume the wave function and energy to be spin independent. The potential V(r) is assumed to be an indepen-

dent function and not to vary. The y,(r) may be sought as an expansion in some

finite set of functions y,(k, r) (not necessarily orthonormalized) satisfying the conditions (4.6.22)

Wale) = Y cy(k) Oylk, F) Here the coefficients c¥(k) and c,(k) we caH the approximate variational that the more functions 9,(k, r) we more exact the method. The various

(4.6.25) are treated as trial parameters. This is what Ritz—Galerkin method. It stands to reason take and the happier the choice made, the band calculation methods differ from each

other in the choice of the set of {g,}. We substitute (4.6.25) into (4.6.24) and

338

4. Band Theory

carry out the variation with respect to c¥(k) (the calculation is similar to that presented in Sect. 4.6.1, but is much simpler). This yields a system of equations

(4.6.26)

Y [uv(k) — ES, .(k)]¢(k) = 0 , where H,(k) = J drop tk, r)H p(k, vr) ,

(4.6.27)

Sy(k) = f drostk, nok.) a

Variation with respect to c,(k) yields a system of equations which differ from (4.6.26) only in complex conjugation. The solvability condition for (4.6.26) has the form

det | #,,(k) — ES,,(k)|| =0

.

(4.6.28)

Numerical solution of (4.6.28) enables us to determine the spectrum E(k).

46.3 The LCAO

Method and Tight-Binding Approximation

Let us try to choose the wave functions of electrons in an atom, x,(r), as the basis functions, similar to the treatment in the one-dimensional model problem

considered in Sect. 4.1.3. Admittedly, they do not satisfy Bloch’s boundary

conditions, but the solution of (4.1.46) obtained in the model problem points to a way out of this difficulty. We introduce the functions

0,(k, 9 = NY

P

y,(r—R,)exp(ik:R,) ,

(4.6.29)

which, as can be readily verified, satisfy Bloch’s theorem and, consequently, the boundary conditions (4.6.22). The functions , are linear combinations of atomic functions, to which the LCAO (linear combinations of atomic orbitals) method owes its name. Substitute (4.6.29) into (4.6.27):

H,,(k) = N~"¥ explik-(R, — R,)] f dryt(r — Ry) "

a

= Vexp(—ik-R,) f dryf(r— R,) a

re

.

7,0)

Su(k) = ¥ f drexp(—ik- R,)x(r — Rx) 7.2

x,(r — R,)

(4.6.30)

4.6

Computing the Electron Energy Spectrum of Crystals

339

Here we have made the replacement of variables in the integral r > r — R, and

have introduced the notation R, = R, — R,.

In principle, if we take a sufficient number of basis functions (4.6.29), we can

obtain a fairly good description of the energy spectrum of the crystal by substituting (4.6.30) into (4.6.28). The wave functions of the continuous spectrum of the atom are, as a matter of fact, not included in the set of the expansion so that the system of functions (4.6.29) is not complete and the accuracy of the method is restricted. Clearly, the LCAO method reproduces the core states—at least the d band—better

than, say, the states of conduction electrons in alkali

metals. This is so because the states of electrons in narrow bands, with a weak overlap of the wave functions on different sites, are largely determined by the atomic states, whereas an ordinary plane wave that would provide a fairly good

description of the state of conduction electrons in metals is very difficult to

construct by use of (4.6.29) one has to take the complete set of atomic states

including the functions of the continuous spectrum of the atom.

On the other hand, as is obvious from Sect. 4.5.3, in narrow energy bands the correlation effects are substantial. Therefore, the wave functions and oneelectron energies thus found should be substituted further into some manyelectron model; i.e., they are related to the observables only indirectly. Neverthe-

less, the approximation concerned has still been used in band calculations, although not as frequently as some of the more sophisticated methods (see

below). In the various model calculations the tight-binding approximation [4.2] is frequently used, in which the basis of the LCAO expansion contains one function: one atomic level, one band. This approximation, strictly speaking, is applicable when the bandwidth is much smaller than the distance to the neighboring atomic level (Fig. 4.34). As a model approximation, it may, however, be employed in a more general case, since it frequently reproduces the character of the spectrum qualitatively correctly. From (4.6.26) we obtain

E(k) = %,,(kK)/S,(k)

(4.6.31)

Fig. 4.34. Derivation of the applicability conditions for the tight-binding approximation

340

4. Band Theory

Normally we assume that S,,(k)~1, ic; the overlap integral of the atomic functions on different sites is neglected:

J dryt(r — Ry) x,(1) © Oyo -

(4.6.32)

a

Substitution of (4.6.30) into (4.6.31) yields

E(k) = &, + 4e,+

Y Byexp(—ik- R,) .

(4.6.33)

nt0O

Here

Bun = fdrxt(r — Ry) H x,(0r) 5

(4.6.34)

"

&, + de, = fdryt(r) # x,(r) , Q

with e, denoting the energy of the atomic level, and 4e, the shift of this level

owing to the variation of the potential in comparison with the atomic potential. The fact that the integration (4.6.34) is not over the entire space but only over the

cell is immaterial, since the falloff radius of y,(r) is small compared with r,; By. are called transfer integrals, which decrease rapidly with increasing n. The nearest-neighbor approximation consists of only the integral 8,, being retained in (4.6.33). By symmetry, this integral for s states (only) depends on |R,| alone—

not on its direction

E,(k) = ¢, + Ae, + By Yexp(— ik’ Ry) ,

(4.6.35)

é

where the integral is summed over the nearest neighbors. Using the expression for the translation vectors in simple lattices (sc, bec, fcc,

see Chap. 1), we obtain (the calculus is left as an exercise) E,(k) = €, + 4e, + 2B,,(cosk,a + cosk,a + cosk,a)

(4.6.36)

for sc lattices,

E,(k) =e, + 4e, + 4B ( cos ka cos 4 + cos K4 2

k,a ka ka x cos pat~5 oa + costi#— cos"s*z )

2

2

( 4.6.37 )

4.6

Computing the Electron Energy Spectrum of Crystals

341

for fcc lattices, and k

(4.6.38)

E,(k) = 8, + de, + 8,1 008-2"cos “24 cos fea

for bcc lattices, a being the edge of the elementary cube. It follows from (4.6.38)

that the bandwidth in the approximation of nearest neighbors with the coordination number z is equal to

AE, = 22|B,1| ,

(4.6.39)

ME, = 16/811. for sc and bcc lattices, and for fcc lattices, respectively. As follows from the exact formula (4.2.20), the band energy represented in the form of (4.6.33) if B is taken to imply some

parameters.

is exactly undefined

We can also introduce relevant wave functions, similar to g,(r), such that

the condition holds

Vaclt) = NY ggc(r)exp(ikRy) ,

(4.6.40)

where y,-(r) is an exact Bloch function. The functions ¢,,(r) are said to be Wannier functions. For narrow bands they resemble atomic functions and, at any rate, exhibit a maximum on the nth site. To find the explicit form of ¢,;(r),

we multiply (4.6.40) by N ~ '/exp(— ik: R,,) and perform a summation over k with respect to the Brillouin zone. Making use of (4.5.3), we obtain

NOMS Wac(rlexp(— ik Ry) = N~*Y, ul) “Lexp [ik -(Ry — Rn)] = ¥

(4641)

acl?) Sam = Pelt) -

In contrast to the atomic functions x,(r — R,,), the Wannier functions are strictly orthonormalized:

fdr oR(r)Ome(r) = No Y f deb (ry vec (r) a kk Q -exp(ik: Ry, — ik’: Ry) = N~'Y exp(ik- Ry kk

= ik! Ry )Oqy-Sce: = SeeN 7? Sexp Lik: (Ry — Rw) k

= b¢- Simm «

(4.6.42)

342

4. Band Theory

Here and henceforth we assume the Bloch function to be normalized to a single cell (Sect. 4.2.3)

Sardi (Wace (”) = Sea dee -

(4.6.43)

Q

Wannier functions are utilized in considering the quasi-classical dynamics of an

electron, localized states, some magnetic phenomena, etc.

4.6.4 The Orthogonalized Plane Waves (OPW) Method. Pseudopotential As stated previously, the LCAO method is of little use for describing the states of conduction electrons in metals. The plane wave (PW) method suits this purpose better. The basis functions chosen in the PW method are as follows:

x(k, r) = exp [i(k + b$)-r]/V5? .

(4.6.44)

The functions x,(k, r) satisfy (4.6.22). The expansion of ,;(r) in (4.6.44) was employed in (4.2.13). The above expansion is convenient in proving the various

properties of the Bloch functions, but is practically unacceptable for real computations because the convergence of the method is very poor; for reasonable results to be obtained, a large number of functions have to be included in

the expansion of ¥,,(r). The point is that, before arriving at the states of conduction electrons, we need to reproduce the ground states w,(k, r), which lie

lower in energy and are completely dissimilar to plane waves. It is far more

convenient to seek the conduction electron states by the LCAO method. Herring [4.38] suggested that orthogonalized plane waves (OPW) be used as the expansion basis

On(ksr) = rulkvr) — Yo Wher) fdr i(k ¢ Q

ra(k re’) -

(4.6.45)

These waves satisfy (4.6.22) and, besides, are orthogonal to all the core states (it is, in a way, a matter of convenience to decide which states exactly should be included in the core states):

f dros (k, r)W.(k,r) =0 .

a

(4.6.46)

Equation (4.6.46) follows immediately from (4.6.45) and the orthonormalization

of the core states.

In the expansion in the functions (4.6.45), the core states cancel out, and we

immediately seek the conduction-electron states, which are already well described by a linear combination of a small number of OPW. The convergence improves drastically.

4.6

The

OPW

method

Computing the Electron Energy Spectrum of Crystals

is rather more

than

merely

a lucky

343

computational

subterfuge. It has given birth to the concept of the pseudopotential, which has

exerted a tremendous influence on the entire solid-state theory [1.31, 4.39, 40]. The main idea of the pseudopotential method is as follows: In many cases we

are interested in a small segment of the energy spectrum near the Fermi energy;

only these states determine the thermodynamics of the metal, its kinetic properties at frequencies that are not too high, phonon spectra, etc. (but not optical or X-ray spectra). An effective Hamiltonian can be selected in which this area of the

spectrum coincides with the exact region (for example, the core states are thrown away, as was done in the OPW

method). The throwing away of a large number

of bound states corresponds to an abrupt decrease in the depth of the potential

well on each site. One may expect that the effective potential (pseudopotential) that has arisen will already be so weak that it can be taken into account using

perturbation theory. this pseudopotential approximation and, scribing conduction

As mentioned in Sect. 4.2.5, the possibility of constructing accounts for the surprising success of the free-electron especially, the nearly-free-electron approximation in deelectrons in metals. The pseudopotential is defined am-

biguously and may be introduced in many ways. One of the possible methods of constructing a pseudopotential is as follows:

Let the Schrédinger equation have a spectrum E,, E,,... and relevant eigenfunctions w,, f2,... . The energy levels of interest are those beginning from

(n+ 1). Try y,, for m > n + 1 in the form

y=o- x, Vn(Wms @) »

(4.6.47)

with (yp, g) being a scalar product. Here

(W,Um)=0,

m=1,2,...,7

(4.6.48)

since (Wx, Wm) = 0 for m # m’ < n. Substituting (4.6.47) into (4.2.4) gives

HO - ¥ (HV uMbmr 9) = ES EVnl hme 0) or

Hap =Eg,

(4.6.49)

B= K+ FS (E-EgWnlWm a

V + V, is the pseudopotential.

= FAVED,» 2

~

(46.50)

344

4. Band Theory

The functions E, 41> E,42)--. can term in (4.6.47, 50) E,, E,,..., E,. The

W,.1,W,+2,--and the corresponding eigenvalues be obtained from (4.6.49), since for y,,(m > n) the second is identically equal to zero. We exclude the eigenvalues pseudoHamiltonian (4.6.50) is nonlocal and, besides, de-

pends on energy, although the latter is immaterial, since we can always add to 3, an arbitrary term of the form

H' = ¥

Valo

2) :

(4.6.51)

where f,, are arbitrary functions of coordinates and energy. This comes from the fact that the operator 2’ has zero matrix elements between any two functions

Wms Vm:(m, m' > n) and, therefore, does not affect the energy spectrum. The nonlocal nature of the pseudopotential means that its matrix elements

between plane waves (k| V + V, |k’) depending not only on the difference k — k’ but also on each k, k’. This fact should be borne in mind in pseudopotential

calculations involving the use of perturbation theory. In practice, the pseudopotential is constructed either analytically or by selecting a model with several

fitting parameters, which are determined from experiment and then used to

interpret a great many other experiments. As follows from the definition of the pseudopotential in (4.6.50), it is a sum of the periodic potential of a real crystal, V(r) in (4.2.4), and some operator V,

which is involved in (4.6.50). The diagonal matrix elements of these summands

are of different sign, negative for V(r) (in the ion core region) and positive for the

operator V,: (W|V,I¥) = Y= 1(E — En) I(G, Wm)I? (Since E > E, for m |), too. Then we may proceed from

the operators K and # to the momentum and coordinate of the wave-packet

center and understand (4.7.29) as an equation for averages. Let us introduce

=K-£ An. he

(4.7.30)

The velocity of an electron then is equal to

_=5LOE CK) _ OE

(4.7.31)

We calculate the rate of change of the kinematic wave vector k. Differentiating (4.7.30) with respect to time and taking into account (4.7.29, 31), we find

4.7

. e k= ~ ha1o Bl Kao]

~£(,

~ fic

2A:_ , 2A;

\" or;

Band Electrons in a Magnetic Field

359

e. Edin (4.7.32)

"4 or; )

or, in vector form,

k =< ~ hie (0x 0 H)

(4.7.33) te

(Lorentz force). The equivalence of (4.7.32, 33) can be readily proved by writing out the components with allowance for (4.7.2). Equations (4.7.31, 33) serve as a basis for describing the classical dynamics of an electron with an arbitrary dispersion relation [i.e., exhibiting the E,(k) dependence].

Finally, let us estimate at what field the magnetic breakdown sets in—i.e., the field at which the probability of interband transitions increases sharply. For the

electric field we had the inequality (4.4.25). Proceeding to the magnetic field, we need to make the following replacement, in keeping with the expression for the

Lorentz force:

hle|H hog le|F + |e|vH/c x med qa where the velocity of an electron at the boundary of the band has been estimated

to be h/md. Using this replacement, we obtain the result that the probability of a magnetic breakdown is small if 2

hoy < 4 ~

2

,

(4.7.34)

with A being the width of the forbidden energy band. For some metals subject to

fields H < 10° Oe = 8- 10° A/m, this inequality may have an opposite sense. For a more detailed treatment of the magnetic breakdown theory see [Ref. 4.8,

Sect. 10].

4.7.2. Classical Paths Our objective now is to solve (4.7.31, 33). Scalar multiplication of (4.7.33) by » yields

ok=0.

(4.7.35)

360

4. Band Theory

Substituting (4.2.38) into (4.7.35), we find

E(k) =0.

(4.7.36)

Consequently, when an electron moves in a magnetic field, its energy does not vary (as already stated); i.e, it is an integral of the motion. Scalar multiplication of (4.7.33) by H gives

Hk=0.

(4.7.37)

That is, the projection of k onto the field direction (z axis) is also an integral of

the motion. Then the equation of the path of an electron in reciprocal space will be

Eclkzs ky, KM) = Eo ,

(4.7.38)

where k‘°) and Ez are fixed (for paths that actually exist, Ey is the Fermi energy), and k, and k, run over all the possible values that are compatible with (4.7.38).

Thus, the path of an electron in reciprocal space is the contour where the isoenergetic surface crosses the plane perpendicular to the field. Hence it is clear that a study of the various properties of metals in a magnetic field provides

valuable information about the shape of the Fermi surface. The various types of isoenergetic surfaces are shown in Fig. 4.37. In analyzing the quasi-classical

dynamics of an electron, it is convenient to exploit the extended-zone method.

The geometric properties of the surfaces portrayed in Fig. 4.37 (single or multiple connectivity, the presence or absence of self-crossing curves, etc.) have a very substantial effect on the character of electron motion in a magnetic field. These problems are discussed in more detail by Lifshitz et al. [4.8]. Here we restrict our attention to the fundamental difference between open and closed

paths. The motion in open paths is infinite, and the motion in closed paths is

Fig. 4.37. Different types of isoenergetic band electron surface

4.7

Band Electrons in a Magnetic Field

361

finite. The possibility of infinite motion in a plane perpendicular to the field distinguishes an electron is a crystal from a free electron. Thus, the motion

in open paths is not quantized, and, as a result, galvanomagnetic effects, particularly the behavior of the magnetoresistance, can vary appreciably [Ref. 4.8, Sect. 28].

Observe that the trajectory of an electron in reciprocal space and the trajectory of an electron in direct space are similar. This follows immediately

from (4.7.33): the electron velocity vector in & space in the plane perpendicular to the field is equal to the velocity vector in r space, when the latter is multiplied by

|e|H/hc and rotated by 90°. In consequence, the counterpart of infinite motion in k space is infinite motion in r space, and vice versa. We express (4.7.33) in terms of the components

_ oH

K- —F,

ke = Fey

”

he

(4.7.39)

We square these equations, combine them, and extract the square root

dk,

lel a

(4.7.40)

where dk, = (dk? + dk?)'/?, v, = (v2 + v2)". Consequently, the period 7, of motion of an electron in a magnetic field in a closed path is

A

=

he \elH?

dk, = 2n vo, wy’

7. (4.7.41)

with the integral being taken over the path (4.7.38). The quantity w, in this

equation is the cyclotron frequency of a band electron; it may be represented in the ordinary form of (3.5.90):

lelH

4 mnlEo, Ye ’

4.7.4 4742)

= ———.-

where, however, the cyclotron mass is a function of Ey and k: h dk my(Eo, ki) = ano ov,

.

(4.7.43)

The integral involved in (4.7.43) can be put into a more convenient form. To

achieve this, we calculate the area spanned, on the one hand, by trajectory A (4.7.38) with the parameters E and k‘ and, on the other, by trajectory B with the

362

4. Band Theory

ky

Ay

Erde

Fig. 4.38. Derivation of conditions for quantization in a magnetic field

parameters E + dE and k® (Fig. 4.38): dS =

j

E 0. Furthermore,

iy =

; mm

m

1/2

3

m, a; +m, 03 +303

;

(4.7.49)

where a; stands for the field direction cosines with respect to the major ellipsoid axes (i.e, the corresponding projection H, = Ha,, H, = Ha, H, = Has). Here

the cyclotron effective mass also is the same for all electrons, but it depends on

the direction of H. Expression (4.7.49) may also be obtained straightforwardly by considering diamagnetic resonance is semiconductors [Ref. 4.50, Chap. 13]. A rule can be deduced from (4.7.33): an electron moves in such a way that a smaller-energy region lies on the right of the direction of motion at each point of the trajectory. For closed paths this signifies, according to (4.7.46), that if

m,(E, k®) > 0, the electron revolves in the same direction as that in which a free electron does, and for my(E, k°) < 0 it revolves in the opposite direction. Thus one talks of electron and hole trajectories. With the k dependence of wy, it might seem unclear at what frequencies cyclotron resonance should occur in metals (insofar as the E dependence of wy goes, it is of course necessary to set

E=E,,

since only thermally

active electrons can

receive energy

externally applied ac field and participate in resonance). frequency is

@=nw, (Ep, kK),

from an

If the external-field

n=1,2,3...,

(4.7.50)

the electrons that contribute to resonance are those for which

Jon(Ep, kp) — Oy(Ep, KO) S77? ,

(4.7.51)

with t being the time between collisions (Sect. 3.7.3). In the general case, such electrons occupy a layer of thickness

| 4k, | (i

Oy Ok,

1 |k,= Ko

,

364

4. Band Theory

But if dw ,(Ep, kz)

ak,

=0,

lk, = ki

(4.7.52)

the number of such electrons is appreciably larger. For them,

|Ak,| & (« Therefore,

i

wy Ok?Zz

k, = KO

resonance

will occur

primarily

when

conditions

(4.7.50,

52) are

fulfilled. Apart from this, absorption anomalies can take place for minimum and maximum values of k®. At these points the velocity is parallel to the magnetic field (Fig. 4.39).

Fig. 4.39. One of the types of points contribute to cyclotron resonance

in k space

that

4.7.3 Quasi-Classical Energy Levels. Oscillatory Effects The motion of an electron in a plane perpendicular to the field can also be finite. Then, according to the general principles of quantum mechanics, only certain values of the corresponding degrees of freedom will be allowed values. We quantize the effective Hamiltonian (4.7.28) according to the Bohr-Sommerfeld rule. Here it is convenient to choose a vector potential (calibration) other than that used in (4.7.3):

A,=—-Hy,

A,=A,=0.

(4.7.53)

The effective Hamiltonian then assumes the form

.

H

Wa = £(K.+ fe y. Ky, K.)

(4.7.54)

4.7

Band Electrons in a Magnetic Field

365

The variables x and z are cyclic, so we have a unique quantization condition

§K,dy = 2n(n+}),

n=0,1,2,...,

(4.7.55)

where the integration is carried out over the closed trajectory in phase space. It

follows from (4.7.53) that K, = k,, and, since x is a cyclic variable, K, = 0. Hence we obtain

k, =

dk,

eH

dy .

he

(4.7.56)

Substituting (4.7.56) into (4.7.55) and allowing for $k, dk,, = S(E, k,), we find 2n\e|H

S(E, k,) =

he

(4.7.57)

(n+4).

Condition (4.7.57) defines the energy-band spectrum E (n, k,). Strictly speaking, this condition (like the quasi-classical approximation) is only valid for n> 1.

The energy difference of two adjacent levels here is equal to AE

_ 2nle|H fic

(AS(E, k,) GE

)

-1

= hwy(E, k,) «

(4.7.58)

(see 4.7.42, 46). Thus, the correspondence principle holds: the frequency of the transition between two adjacent levels at large quantum numbers is equal to the

rotation frequency. Note that the above standard derivation of (4.7.55) applies here only to a

Schrédinger equation of second order in @/dy. Nevertheless, it also works well in the more general case [4.51, 52].

Assuming hw, to depend weakly on E and k, (the dispersion relation is close

to the quadratic law), the result for the number of levels under the Fermi surface

will be

ng © Ep [hoy «

(4.7.59)

The condition n, > 1 is equivalent to (4.7.21) if kp + d~'. With nearly filled or nearly empty bands it may turn out that ng = 1 and (4.7.57) should be inapplicable. However, the dispersion relation here is close to the quadratic

law and the effective Hamiltonian is a harmonic-oscillator and free-motion Hamiltonian. For a harmonic oscillator the quasi-classical spectrum is known to coincide

with

the exact

spectrum

even

when

n= 0.

Thus,

quantization

condition (4.7.59) may be regarded as applicable practically without restrictions (except for those required for the effective Hamiltonian itself to be applicable, Sect. 4.7.1).

366

4. Band Theory

The energy levels do not depend on the integral of motion K,. We calculate the corresponding degeneracy multiplicity. The number of states with a given

spin projection k, that lie in the dk, interval, with n being fixed, is defined as

Vdk, AS/(2n)*, where V is the volume of the crystal and AS the difference of the areas in the k,, k, plane that correspond to the levels E, and E,,,. Taking into

account (4.7.57), we obtain for the degeneracy multiplicity of each level

dq(n,k,.) V lelH_ OV dk, (2x)? hc (2nly)’

(4.7.60)

as for free electrons (3.5.105). As can be seen, the degeneracy multiplicity is independent of n and k,.

As stated in Sect. 3.5.6, quantization of the motion in the x, y plane leads to characteristic oscillations of the various physical quantities as a function of H ~!

(for example, magnetic susceptibility, the de Haas—van Alphen effect, Sect. 3.5.6). The density of states at the Fermi level exhibits singularities when a successive Landau quantum level passes through the Fermi surface. These peculiarities, leading to oscillations, manifest themselves if the distance between adjacent Landau levels is larger than the thermal smearing of the Fermi level and the broadening of the levels themselves, which is due to collisions,

hoy 2 kT, h/z .

(4.7.61)

Inequalities (4.7.61) can be fulfilled at low temperatures (on the order of several

K) in pure samples and high magnetic fields (H < 10° Oe x 8- 10° A/m). The

oscillation period can be determined directly from (4.7.57)

A(V/H)=

2n\e|

1

SiEek)

(4.7.62)

since it is for this magnetic field variation that the number of the Landau level at

the Fermi surface varies by unity. Just as in the cyclotron resonance case, we may ask exactly what k, values contribute to the oscillation period. For the same reasons as those for cyclotron resonance, the contribution is made by the intersection of the Fermi surface with

the extremal area (Fig. 4.40). In any other case the oscillation periods due to the intersection of k, and k, + Ak, differ considerably, and such oscillations, on the

average, suppress each other. But if 0S/ék, = 0, these periods differ by an

Gy

Fig. 4.40. Extreme cross sections of the Fermi surface

4.7

Band Electrons in a Magnetic Field

367

amount «(4k,)’, ie. the oscillations in the vicinity of this point occur “in phase” and enhance each other. Thus, investigating the oscillation period of the susceptibility or magnetic moment as a function of H~! enables us to determine the areas of the extreme intersections of the Fermi surface in any direction.

Important information is also contained in the magnitudes of the oscillation amplitudes. In simple cases, when the Fermi surface has a center of symmetry

and any arm drawn from the center meets the surface only at one point, the form of the Fermi surface can be unambiguously reproduced from a knowledge of the oscillation periods, at arbitrary field directions. The temperature dependence of

these amplitudes enables us to find the velocity distribution at the Fermi surface

[4.53]. Broadly speaking—i.e., for a Fermi surface of arbitrary shape—a study

of the de Haas—van Alphen effect is insufficient for an unambiguous determi-

nation of the energy spectrum of electrons near the Fermi surface. For this purpose we also make use of cyclotron resonance, the anomalous skin effect, ultrasonic measurements, so-called dimensional effects in thin metallic plates placed in a magnetic field, studies of galvanomagnetic properties, and positron annihilation. We do not linger on these very important problems; but they are elaborately treated in the literature [4.8, 44, 54]. Here we note that the initial approximation normally used in processing experimental data is a Fermi surface constructed in the nearly-free-electron approximation (Sect. 4.6.4). We conclude this section by mentioning the very interesting phenomenon of

first-order magnetic phase transitions under conditions of strong susceptibility

oscillations (Shoenberg effect, see [4.8, 55]). The point is that we must take into account the magnetic field that acts on the electron not only due to the external

field but also due to the other charges in the metal. Since the characteristic scale of the electron orbit r,, is large compared to the lattice period, the electron senses the medium’s magnetic field, which is produced by all the external and internal

charges. The averaged intensity of a field in a medium is the magnetic induction

B. Thus, the electron-electron interaction in the de Haas—van Alphen effect can be adequately taken into account by the replacement H — B in the formula for

the magnetization M(H). At sufficiently low temperatures the M(H) dependence

may be very strong (this is clearly manifest in the simple two-dimensional model of the de Haas—van Alphen effect considered in Sect. 3.5.6). Consequently, upon

performing the replacement H — B, it is the M(B) dependence that will be highly pronounced. This may result in the H(B) = B— 4nM(B) ceasing to be a monotonic function (Fig. 4.41). the

The detailed theory of the de Haas—van Alphen effect shows that at 0 K in absence

of scattering

processes

the

amplitude

of the

oscillations

of

M(B) x B~'/? and 0H/dB = 1—4n06M/0B is bound to become less than zero when form

the B’s are sufficiently small.

ng & (c/vp)*? = 10° ,

The corresponding

condition

has the

(4.7.63)

368

4. Band Theory

Fig. 4.41. Derivation of the Shoenberg effect

with n, being the number of Landau levels beneath the Fermi surface. On the other hand, the thermodynamic equilibrium conditions for magnets require that the inequality

0B (Sa).

>0

(4.7.64)

be fulfilled [Ref. 4.50, Chap. 5]. Therefore, the BC segment in Fig. 4.41 corresponds to a thermodynamically unstable state. As is always the case in such

situations, the true H(B) dependence is obtained by means of the Maxwell construction [1.4]. The horizontal straight line AD should be drawn in such a fashion that the surface area of ABO is equal to that of OCD. The H(B) relation

thus constructed describes the first-order phase transition between states with

different values of H and with the same value of B. The straight line AD is a phase coexistence line, the segments AB and CD describe a metastable state, and

the segment BC describes an unstable state. This Shoenberg effect leads to

alterations in the observed character of the oscillations, and also gives rise to a peculiar domain structure; i.e., it results in the metal being broken up into sections with different magnetization.

48 Impurity States

48.1 A Simple Model Departure from ideal spatial periodicity may, just as with phonons (Sect. 2.4), lead to the occurrence of new states which are localized at a distortion-inducing

defect. We restrict ourselves to point defects (impurity, vacancy, primary objective now is to consider a simple model problem.

etc.). Our

48

Impurity States

369

Let there be a lattice of atoms, on each of which the electron is in one orbital

state (tight-binding approximation, Sect. 4.6.3). The energy of the atomic level on one of the sites (we choose that site as the origin) differs from those on the other sites by U. Then the Schrédinger equation becomes

¥) Bandmt USi040 » Bam =O , Ea,= mén

(4.8.1)

where a,, is the amplitude of the probability that the electron will be located on

the mth site, and 8,,, is a transfer integral, that is, the matrix element of the

Hamiltonian between the states of sites m, n (for simplicity, we assume that it depends only on the difference R, — R,,; i.c., the case in which one of the sites is a zero site is not singled out in the B,,,). The energy E is counted from the atomic-

level energy of the “host” atoms.

The function a,, is defined on a three-dimensional lattice R,, and may always

be represented as a Fourier integral

ay, = I ¥ a(k)exp(ik-R,,) , NY

(4.8.2)

where the summation is carried out over an arbitrary reciprocal-lattice cell (for

example, over the first Brillouin zone), and N is the number of lattice sites. Substituting (4.8.2) into (4.8.1) and using the Fourier integral expansion of B,,,,

Ban = yj & Blades —ia-(Ry— Ry) «

(483)

we obtain

EE, ahyexp(k- Ra) = 75, Alaa)exp ilg—H)- Ry -exp(ig: R,) + . y a(q) x exp(ik-R,) ,

[6 = x Dex

(4.8.4)

R)| .

Allowing for Rn] = Nou Yexp[—-i(g—A)-

(4.8.5)

370

4. Band Theory

and equating the coefficients of the orthogonal functions exp(ik: R,), we have U

CE— Blk) alk) =

2 a(q) . a

(4.8.6)

Two types of solutions exist. In the first, )., a(¢) = 0, i.¢., ag = 0. The probability of finding an electron on the impurity is equal to zero, the spectrum coinciding with that of the band states, E = B(k). In the second solution, ay # 0. Then (4.8.6) yields U

1

(4.8.7)

I= ERB

When min B(k) < E < max B(k) the question arises as to what the singular integral involved in (4.8.7) should imply. As with phonons, it is solved by introducing a small imaginary addition to E (Sects. 2.4, 6). Such states describe band electron scattering on an impurity.

case

We defer the problem of describing these states for a while and consider the E < min B(k)

(4.8.8)

E> max B(k) .

(4.8.9)

or

It follows from (4.8.7) that (4.8.8) may hold for U 0. Further, for illustration, we consider the case of U 0), and represent (4.8.7) as

|U| 1 =! NY 44h) =|U|F(4).

( 48.11 )

with B(k) = B(k)— min B(k). Assume without loss of generality that the minimum of B(k) is reached at the point k=0 (if necessary, we displace the

boundaries of the cell in reciprocal space; the case of several equal minima does

not present any difficulties, either). At small & h?

;

Mes

3|%

Bik)=> y

(4.8.12)

48

Impurity States

371

where d is the space dimension (d = 1, 2, 3) and m, is the eigenvalue of the inverse-effective-mass tensor (all m, > 0). In the one- and two-dimensional cases,

the value

1

1

dk

1

10 WE Fay” en Fa

(4.8.13)

(with V, being the unit-cell volume) diverges at small k (logarithmically for d = 2

and o k~* for d = 1). Since F(A) decreases with increasing 4 and F(0) = 00, (4.8.11) here can be solved for any |U|. Such states are said to be localized. To understand the meaning of this term, we calculate the expression

aq = N'Y, a(k)exptik- Ry) =e 5

Re

(4.8.14)

where (4.8.2, 6) are taken into account. For 4 > 0, the quantity E — B(k) does not

go to zero anywhere and, therefore, according to the familiar properties of the Fourier transformation, a,, decreases at |R,,|—> 00 more rapidly than any power of |R,,|~! [for real B(&) it decreases exponentially]. The assertion for d=1 constitutes the Riemann—Lebesgue lemma [Ref. 4.17, Sect. 3.1]. The generalization for the many-dimensional case, at least at the “physical” level of rigor, is

trivial enough. Consequently, the probability of detecting an electron in localized states decreases rapidly with increase in the distance from the impurity.

In the one- and two-dimensional cases localized states always arise. In the three-dimensional case they occur when the | U| is sufficiently large, i.c.. when

Iu

U|>|

li

1 —y

1]! x—

FF

|

;

(

4.8.15

as follows from (4.8.11) and from the decrease of F(4) with increasing 4. Localized states with E < min B(k) in our model correspond to donor levels

in the theory of semiconductors, and with E > max B(k) to acceptor levels.

48.2 Green’s Functions and the Density of States We

now

wish to consider the general formulation

Hamiltonian have the form

H=H,+V,

of the problem.

Let our

(4.8.16)

with Ho being the Hamiltonian of an ideal crystal, and V the perturbation. In the one-band (tight-binding) approximation, the V is given by the matrix

372

4. Band Theory

elements

V,,,, where m and n stand for site numbers. Thus, it was assumed

Sect. 4.8.1 that

Vian = U Sino Sno

in

(4.8.17)

We wish to determine the spectrum of the Hamiltonian #, or, to be more exact, the density of states

g(E) = 5 6(E—E,)=Tr{E—#}

,

(4.8.18)

with E, being the eigenvalues of the Hamiltonian #. The operator 6(E — #) is related to the operators

R*(E)=(E—# +in)"|,. +0

(4.8.19)

by the identity 1 x+in

n> +0

=7 t F ind(x) ,

(4.8.20)

which should be understood as follows: For any “sufficiently good” function

9(x),

§ dxp(x)(x tin)! =f dx p(x)x(x? +7)! Fin f dxe(x)

(4.8.21)

(x? +197)"! ——> PJ dxo(x)x"! Fing(0) , where

is the symbol of the principal value and where we have employed one of

the representations of the 6 function

" 6(x)= lim : —U_ . (x) a2 00(x? +17)

(

4.8.22

)

The operators Rt (E) are called resolvents, or Green’s functions. The density of

states can be expressed in terms of these R*(E):

g(E) = ¥, (E-E,) = Fim] 5 (e~E,+in

b= ¥ -ImTr{ A*(E)} . (4.8.23)

Suppose that we know the unperturbed Green’s function

RE(E)=(E—Hy +i)!

(4.8.24)

48

Impurity States

373

For this function to be constructed, we need to know the eigenfunctions and the eigenvalues of the Hamiltonian #9. Equations (4.8.16, 19, 23) yield

[R*(E))-' =(R3 (BE)! V=(RE (EN)

(1- RG (E)V) .

(4.8.25)

Transforming (4.8.25), we find R*(E)=[1—R2(E)V]“! R2(E) .

(4.8.26)

We need to prove the matrix equation

detA =exp(Tr{inA}) .

(4.8.27)

Denoting Ind = X, we obtain

det A = det expX = lim aei(1 +*) ao

= lim[148]

n

= lim [ee(1 +*)] n> ©

n

=expTrX ,

which proves (4.8.27). The properties used here are

det(AB)=detAdetB , det(1+eX)=1+eTrX + O(c?) .

We now transform Tr R*(E). It follows from (4.8.19) that (0/E)InR*(E) = —R+(E) and

.

é

>

é

-,

5

@

Tr{R*(E)} = — gp Tr{inR*(E)} = 3p irin [1—Ré(E)V] —3pT

“Ing (E) =A. Trin(l —Ra(E)V]+TrRt(E),

(4.8.28)

where we have used (4.8.26, 27). The result for the density of states, according to

(4.8.23), then is

g(E) = go(E) F Lim n

Z-indet(l —R2(E)ri} . OE

(4.8.29)

Formula (4.8.29) gives a formal solution to the problem of the variation of the

density of states due to the perturbation V.

374

4. Band Theory

We apply (4.8.28) to the problem considered in Sect. 4.7.1. The eigenfunc-

tions and the eigenvalues of the Hamiltonian 9, have the form

a,(k) =N-*exp(ik-R,) ,

E(k) =B(k) ,

(4.8.30)

where & runs through the first Brillouin zone. Then

[Ro (E) Inn = y am(k)a,(&)[E — B(k)— in]!

(4.8.31)

= N~*Y expik-(Ry— Ru)ILE—B(k)— in}? Using (4.8.17, 31), we find the matrix 1— R (E)V [1 — Ro (E)V Inn = San— UL RG (EV mo Sno +

(4.8.32)

This matrix differs from the unit matrix only in that it has a zero row. Therefore

det[1—Ro(E)V]=1-—UF(E) ,

(4.8.33)

where

F(E)=[Ro (E)]Joo = NN! LE

Btk)— in}?

(4.8.34)

Then

g(E) = go(E) +

1a

Im{In[1 —UF(E)]}

== 9o(E)— nT UF(E)’ golE) Lm)

(4.8.35)

“py = FE) F(E)=— If E>max B(k)

or E = NY CR(E, k)> exp[ik-(R,—Rm)J -

(4.9.2)

If the spectrum is determined as the poles E(k) of the function < R(E, k)), it depends on quasimomentum, as in the case of an ideal crystal. But since the

380

4. Band Theory

mean resolvent is no longer the resolvent of some Hermitian operator, the poles of the resolvent, as will be seen, are not real:

E = e(k) + il (k)

(4.9.3)

(with e and I being real). This fact, broadly speaking, signifies that the energy

spectrum of a disordered system cannot be specified by the dispersion relation

E(k). Physically, this is due to the nonconservation of quasimomentum when the impurity potential is scattered by fluctuations. Recall that the system is assumed to be homogeneous only on the average, but not for each individual impurity.

The quasimomentum thus is not a “good” quantum number and a

state with a

definite & cannot correspond to a definite energy, but is a wave packet and has

an energy spread on the order of I. As is known from quantum mechanics

[1.12], the occurrence of an imaginary part in the energy formally denotes that the state is nonstationary, its lifetime being

t(k) © h/T(k) .

(4.9.4)

Now we can answer the question about the possibility of describing the electronic structure of alloys with the help of the dispersion relation E(k). Evidently, such a description is reasonable if the real part of the pole of the mean resolvent e(k) is large compared to its imaginary part ['(k) or, put another way, the decay (damping) of states with a definite & is comparatively small. This only

takes place provided that the disorder is relatively weak. It must also be emphasized that this approach obviously does not allow us to consider localized States (Sect. 4.5.4), which cannot, even approximately, be specified by quasi-

momenta in the one-impurity limit. Particular methods of computing the average Green’s function will be considered using the example of diagonal disorder models, which, in simplified form, describe substitutional alloys (Sect. 1.10). We assume that the energies of the atomic levels are independently distributed random quantities and the transfer integrals 8,,, are not random but conserve the same values as those in a

perfect crystal. Therefore Ven = U,5inn >

(4.9.5)

with all U, being distributed with the same probability density P(U,), normalized to unity

j dUP(U)=1.

-@o

(4.9.6)

Thus, in the model of a binary alloy composed of A and B atoms with energies €,

and €, and concentrations 1 —c and c, respectively, the quantity U may assume

49

The Electronic Structure of Disordered Systems

381

two values €, and &, with the probabilities 1—c and c: P(U) = (1—c)6(U — £4) + cd(U — eg) .

(4.9.7)

The average Green’s function in such a relatively realistic model

cannot

be

calculated exactly; approximate computational methods will be outlined later in the text. Here we consider the Lloyd model [4.57], in which

P(U)= r ! n r?4+(U-E,)*

(4.9.8)

©

This model is unrealistic since P(U) falls off too slowly when U + + o; ie., large potential fluctuations are more than probable. On the other hand, the model enables us to calculate the function (R(E,k)> exactly and thus to demonstrate its general properties. Before passing on to particular calculations, we wish to derive a very helpful

representation for the disordered Green's function in the diagonal disorder

model (4.9.5).

.

Let the zero-approximation Hamiltonian 3%) be chosen as the sum of the

transfer Hamiltonian, i.e., the matrix ,,, and the perturbation matrix V in which U,, is replaced by zero (i.e., the perturbation on the nth site is eliminated). The perturbation matrix then has the form

(4.9.9)

mn Sin Val = US We use the identity (4.8.26)

(4.9.10)

,

R=(1- RV)R

where R" is the resolvent corresponding to the zero approximation. Also, we use the identity

(1-A)"4=14+4(1—A)"! , which

may

becomes

be proved

R=ROFROTOR

by postmultiplying

5

(4.9.11) by (1 —A).

Equation

TMSV OL—ROV OM)? |

(4.9.10) then

(4.9.12)

The operators, similar to T, play an important role in the theory of alloys. For (4.9.9), in which V™ has a unique nonzero matrix element, it is readily

understood that 7) will also have a unique nonzero matrix element with the same row and column numbers. This may be shown by expanding (4.9.12) in a power series of V. The second-order term, for example, is equal to

382

4. Band Theory

(VOR

Vv),

= Y VO ROY

” =

ie. the operator V number R

(4.8.33).

(4.9.13)

UZ Yb np 5 pq5qtOmnOinR

pq

=

UZ ROS mnDin

>

is replaced by the number U,, and the operator R

by the

As a result,

Tat = TrOamdat »

(49.14)

@)

T, = U,/(1—U, Re)

.

Formulas (4.9.12, 14) permit separation of the explicit U, dependence of the operator R, since R™ is independent of U,.

Because the quantities U,, at different m are distributed independently, expression (4.9.14) makes it possible to average over the potential of a fixed site n

with the other potential being arbitrary a) =

c

j

—o

dUPU)

U Rw

(4.9.15)

.

In the Lloyd model (4.9.8) the integral (4.9.15) is easy to calculate in the

complex plane. The imaginary part of the function R¢® is positive, since it is proportional to the density of states (recall that R= R~). Therefore, the function 1 — UR“ does not go to zero for Im U > 0, and the only singularity

that remains in the upper half plane when the path of integration is closed is the pole

of the function

Therefore

P(U)

Egt+il oe () — 1—(E,+iF)R®

at point

U = E,+il

°

with

the

residue

of (2zi).

(4.9.16)

and the averaging over U, in the Lloyd model reduces to the replacement U,, + Ey + il’. This being valid for any n, we arrive at the result that the average Green’s function in the Lloyd model coincides with the Green’s function of a

system with a nonrandom complex perturbation, equal to Ey + iI on each site, regardless of n. This permanent perturbation may be taken into account by the addition to the energy

1

CR(E, k)) = Ro—(E Eo il’, k) = Fi g 0

(4.9.17)

4.9

The Electronic Structure of Disordered Systems

383

Then the poles of the mean resolvent are equal to E=E

+ B(k)+il ,

(4.9.18)

which tallies exactly with (4.9.3). Only for ['—+0, when, from (4.9.8), P(U) — 6(U — Eg), are the poles of the Green’s function real. Allowance for potential fluctuations entails a damping of electronic states. In the Lloyd model we have come to the result that this damping is independent of E.

4.9.2 Approximate Methods of Computing the Average Green’s Function in the Binary Alloy Model

In this section we consider a more realistic model with the probability distribu-

tion (4.9.7). Exact averaging is here no longer possible, and we have to employ

approximate methods of calculating < R(E, k)>. These methods, treated here within the framework of the simple model discussed, are used, with some alterations, also in real-band calculations of alloys [4.58, 59]. The approximations that are discussed in what follows may be viewed as resulting from the partial summation of perturbation-theory series over some small parameter. This is normally either the spread in values of the potential &,—&, or the concentration of one of the constituents (c = RamE) + Y Ry E)< 5¥;> RigE) + F Ru(E)Rig(E)Rym(E)CSVIV,) +

(4.9.26)

4.9

The Electronic Structure of Disordered Systems

385

But since the quantities U at different | are distributed independently, we have

N

& ——_| E— Bk) —U__—i0 ”

(49.30) “

[see (4.8.34)]. Proceeding to the Fourier representation (4.9.2) and using the convolution theorem, according to which

D exp[ik-(R,— Rn)] » Ry(E)Rig(E) = R?(E, k) , we find

= R(E, k)+c(1—0)

4? F(E)R7(E, kK) +... .

(4.9.31)

Now werecall the Dyson equation (4.9.19) and regard (4.9.31) as an expansion of the function R(E, k) as a power series in Z(E, k):

1

CR(E, k)> = E-Bq)_U~

1

aca)

1

2

ts

(1 — 6) 4° F(E)

(4.9.32)

~ E—B(k)—U —c(1—c)42F(E) | Thus, in the second-order of perturbation in A, we have

2,(E, k) = c(1 —c) A? F(E) >

c(1—c)A?

N

YI

1

B@ a0

(4.9.33)

where we have inserted the first approximation E = B(k)+ U into F(E) near the

386

4. Band Theory

pole . The function (4.9.33) has an imaginary part (Sect. 4.8.2):

1 = 7C(1 — €)4? © go( B(k)) :

r(k) =ImyY, (E, Ale pays In

the vicinity

of the

band

edge

B(k) = min B(k) + e(k)

(4.9.34)

(here

effective-mass approximation e(k) = h?k?/2m*, we have (Sect. 3.5) m*(2m*e)!/2V, 1 N 9o(B(k)) = ah

«-0)

in the

(4.9.35)

with V, being the unit-cell volume; recall that go(é) here is the density of states

for one spin projection. Thus, near the band edge, the damping goes to zero as e!/? and will necessarily become larger than the energy ¢ if ¢ is small. This will take place subject to the condition that

c(1 —c)42m*(2m*e)!/?

~—anisSC*O

SES

ie.

e

(4.9.58)

is the largest. Determining the maximum of the exponent in (4.9.58) from AE, we find, to within numerical factors,

a

AE, & (: kpr)

4

a

(4.9.59)

(Observe that at sufficiently low temperatures the quantity 4E, is really larger than kg 7.) Accordingly, the “optimum hop” probability is

4.10

Wo X exp| - ()

1/4

| ,»

Conclusion. The Role of Many-Particle Effects

Ty =const ina

(2)

3

.

393

(4.9.60)

The static conductivity, determined at low temperatures by these optimum hops,

is proportional to wo (the Mott 7 '/* law). For two-dimensional disordered

systems, as can be readily verified, the 7 '/* law is replaced by the “7 '/> law.” The consideration of disordered systems either here or in Sects. 1.10, 2.5, and

4.5.4 is far from complete. This area of physics is currently developing rapidly and many concepts are being revised. Specifically, electron-electron interaction effects appear to be highly important in disordered systems. However, studies of these effects have not yet yielded conclusive results. An examination of these effects calls for very complicated mathematics, and therefore we do not consider them here.

4.10 Conclusion. The Role of Many-Particle Effects

Currently, band theory forms the foundation of the entire electronic theory of metals and semiconductors, permitting a large body of experimental evidence to be systematized and accounted for. Particularly important is the fact that sufficiently reliable techniques exist not only for calculating the energy spectrum (special mention should be made of the pseudopotential method for metals) but also for determining straightforwardly the shape of the Fermi surface and the

velocity distribution on it from experimental data.

The big success of band theory, however, seems surprising at first sight. It is hard to understand why allowing for the electron-electron interaction in the

self-consistent field approximation—which, generally speaking, is unfounded for values of the parameters characteristic of solids—proves to be sufficient. (In-

cidentally, the success of band theory should not be overestimated, as often

happens. In describing compounds of d and f transition metals, it is necessary to take account of the correlation explicitly—Sect. 4.5.3.) As already noted, the Pauli exclusion principle plays a decisive role here. Consider a weakly interacting electron gas. In the absence of interactions the ground state is an electron-occupied region of k space bounded by the Fermi surface. Any interaction will smear the electronic states, giving rise to transitions between them. For pairwise collisions the energy conservation principle should hold:

E,+E,=E,+E, .

(4.10.1)

Here the initial states E, , at the zero of temperature (7 = 0 K) should reside beneath the Fermi surface, and the final states above it, for the Pauli exclusion principle forbids scattering to occupied states. This evidently contradicts

394

4. Band Theory

(4.10.1). Consequently, when T = 0 K, the pairwise collisions are suppressed and the Fermi surface remains well defined. Simple reasoning shows that the decay

of the state with energy E is proportional to (E — E,)’; i.e., it is on the order of T ? for thermally active electrons, and is small irrespective of whether or not the interaction is weak. However, care must be exercised here; the reasoning offered in Sect. 4.5.3 shows that a sufficiently strong interaction is nevertheless capable of destroying the Fermi surface and transforming a metal into a semiconductor.

What we imply when we talk of an appreciable interaction is one that is still less than the critical value at which the Mott transition occurs.

Current carriers in metals are, of course, not quite Bloch electrons; as already stated, they are surrounded by a cloud of other electrons, with which they interact, and also by a cloud of phonons. These carriers are quasiparticles, i.e., excitations of the entire many-electron system, which, however, in many ways resemble Bloch electrons. The properties of such quasiparticles are described by

Landau’s

Fermi-liquid

theory (Sect. 5.2). One

of its predictions is that the

interaction affects the static properties (heat capacity, magnetic susceptibility, etc.) only through parameter renormalization (e.g., of effective mass or magnetic moment). An essential feature here is that the Fermi surface, determined, say,

from the de Haas—van Alphen effect, is the Fermi surface of quasiparticles and,

therefore, to some extent, its shape takes the interaction into account exactly. Many-particle theory shows that the volume bounded by the Fermi surface does not vary when the interaction is included (Landau—Luttinger theorem). The Fermi surface in this case is defined as a surface in &k space on which the electron distribution function undergoes a discontinuity at T = 0K. The discontinuity persists in the many-particle system, although the magnitude of this jump is less than unity [4.64]. Figure 4.45 shows the form of the n(k) function for the isotropic case, in which n(k) depends on |k| only. As before, the quantity A, is determined by the total density n according to (3.5.14), ie.,

kp = (3n?n)'?

(4.10.2)

Fig. 4.45. Change

of the

Fermi-Dirac

distribution

function A, for the mean number of occupied states with allowance made for the electron-electron correlation

We now wish to “prove” that the lifetime of the band state is long and the uncertainty of its energy therefore is small. In writing (4.10.1), we have already pointed out that each electron from the thermal smearing region ~ kgT does

not collide with all n electrons of the system, but only with

part of them,

4.10

Conclusion. The Role of Many-Particle Effects

395

ny @ (kg T/C)n. Apart from this, not all of these collisions are possible, even if

allowed by energy and momentum conservation. Indeed, if the change in energy

during collision exceeds k, 7, there are no vacant sites for an electron that has

lost this energy. Thus the Pauli exclusion principle introduces an extra factor

into the scattering probability = ky 7/{,. Therefore, the scattering probability amounts to ~ (kg 7/f,)? and the lifetime z of the state is = = (kg T/C)? (h/Co), a value which is large even at room temperature. In addition, the energy un-

certainty relation 4E ~h/z yields a very small quantity for AE; this circumstance allows us to view a Bloch electron as a stable quasiparticle.

We are now at the end of our discussion of the part of solid-state theory

based on the postulate that the interaction between electrons is absent, although we have in fact taken it into account to a certain extent in the self-consistent field approximation. In the following chapter, we present a more-or-less detailed

analysis of the consequences of abandoning this postulate.

5. Many-Particle Effects

This chapter deals with electron—electron and electron-phonon interaction effects. Extensive treat-

ment is given to the plasma model of the metal, the Fermi-liquid theory of Landau, polaron theory, and other problems. Superconductivity, excitons, and magnetism are also discussed at some length. The exposition employs a minimum of mathematical machinery and will serve as an introduction to the physical ideas of contemporary quantum many-particle theory.

5.1 Plasma Phenomena. Screening 5.1.1 A Discussion of the Model In this chapter we investigate the theory of the effects in which the electron

interaction is a decisive factor. Of supreme importance are plasma phenomena, first investigated in terms of the quantum theory by Bohm and Pines in 1953 [(5.1-3].

A plasma is an ionized gas. The presence of a large number of free charges bound by long-range Coulomb forces means that many properties of the plasma

differ drastically from those of a normal gas (for example, the behavior in a

magnetic field and screening). The gaseous plasma has been intensively investigated in connection with the problem of controlled thermonuclear fusion, as well as in astrophysical and geophysical problems (the ionosphere, the physics of the sun, etc.).

On the other hand, the definitive property of the plasma, viz., the presence of free charges, is also inherent in metals and semiconductors. Despite the quite

dissimilar values of the parameters (electron density, temperature), the free

electrons in a solid behave in many ways like a highly ionized gas. Thus, the electromagnetic waves in metals—plasmons, helicons, etc. (Sect. 3.7.4)—have counterparts in the gaseous plasma where they were first discovered. Sometimes

we can ignore the lattice structure and the electronic arrangement of ion cores and regard the metal as an electron plasma with a uniformly smeared positive

ion charge that assures electroneutrality. This is the plasma model of a metal, or the “jellium model”. The model strongly emphasizes the fact that most properties of metals, even purely lattice properties (for example, the magnitudes of

elastic moduli), are determined by free electrons. Frequently this approach turns

5.1.

Plasma Phenomena. Screening

397

out to be very profound, although, like any other model, it has only a certain

scope of applicability. A presentation of the plasma model of the metal will enable us to approach some general and essential concepts and postulates, for example, the fundamentals of linear-response theory.

Different

problems

are

associated

with

plasma

phenomena

in

semi-

conductors; these phenomena are in many respects closer to a “classical” gaseous plasma than to the plasma in metals. First, because of the small concentration, the current carriers in semiconductors may be described by classical Maxwell—Boltzmann statistics. Second, strong electric fields are possible in semiconductors and some nonlinear plasma phenomena occur under those conditions. Although interesting, they do not have a general “solid-state” significance and are therefore not considered here.

The plasma model is exploited chiefly to describe electromagnetic properties

of metals. From the standpoint of electrodynamics, a major property of plasma is its high spatial dispersion, i.e., the dependence of the electric induction at a given point in space on the field intensity in some region. This nonlocality arises

from the long-range behavior of Coulomb forces and shows up dramatically in screening phenomena. To gain an insight into the properties of the plasma, we must consider the response of the electronic system to a spatially varying and time-dependent field.

5.1.2 The Equation for a Self-Consistent Plasma Potential Let the electrons be subject to an external perturbation U(r, t), which will be

regarded as weak and taken into account to the lowest order of perturbation

theory. The perturbation induces in the system a charge-density redistribution which, in turn, results in an extra perturbing potential. The total potential V(r, t)

can be represented as a linear functional of U(r, t) (the linearity here is due to the smallness of U): t

Vit) =fde' f dre“ (nr'yt—r)UCr', t+) .

(5.1.1)

The operator é~' involving the kernel e~'(r, r’; t—t’) is said to be the inverse permittivity (more precisely, the longitudinal permittivity), or the inverse dielectric function. The integration over t’ is carried out to t, since, by the causality principle, U(t) may depend on U(t’) at preceding rather than subsequent

instants of time. Representing U, V as Fourier integrals, for example, Vir.

=

V(r,w)=

i de pier) Sx 21 a

Vir, w) ,

§ dtexp(iwt) Vir, 1) ,

- 2

(5.1.2)

398

5. Many-Particle Effects

we obtain, subsequent to evident transformations,

V(r, w) = fdr'e~'(r,r';@) U(r’, w) ,

(5.1.3)

where

éel(r,r'3@) = t dtexp(iwt)e~'(r, r';t) .

(5.1.4)

oO

For the spatially homogeneous case the quantity ¢~ '(r, r’'; w) depends only on the difference r—r’. Then, performing the Fourier transformation with respect to the spatial coordinates, we find

V(q, w) = U(q, w)/e(g, ) ,

(5.1.5)

where

V(q, w) = § dr exp(—ig-r) V(r, @) .

(5.1.6)

The same applies to U; the 1/e(g, w) is a Fourier transform of e~!(r—r’, w).

In what follows we present a general, although formal, expression for 1/e(g, @) and prove a number of important equations. At the outset it is, however, reasonable to calculate this quantity with comparatively simple ap-

proximations, which actually embrace nearly all of the major plasma phenomena and describe them, on the whole, correctly. To do this, we regard V(r, t) asa

one-electron potential that acts on one-particle states. Using this potential, we then find the self-consistence requirements and thereby take into account its many-electron behavior. This approach for a classical plasma was proposed by Vlasov [5.4]. The Bohm-Pines approach, seemingly based on quite different ideas and called the random-phase approximation (RPA), is strictly equivalent

to the quantum generalization of Vlasov's method. We examine an electron system that is described in a one-particle approxi-

mation by eigenfunctions ,(r) and eigenvalues E, (for solids |v> = |kfo)>, with ¢ being the band number, k the wave vector, and o the spin projection). Now we

introduce a one-particle density matrix

6=Lw,ly> = Ywyly> =P ww (Nfdr'vrrjotr’) .

(5.1.7)

5.1.

Plasma Phenomena. Screening

399

The quantity w, here is the probability of the electron being in the vth state; in equilibrium the w, = n(E,) coincides with the Fermi-Dirac average for any one-particle operator A then is

distribution. The

=Lw j dk. With allowance for

far”

tr

- a exp(ifir),

(5.1.23a)

§ drexp(if-r) = (22)5(f)

(5.1.23b)

a little manipulation yields dq

e(r, r'3 w) = Som?; e(q, w)exp[ig-(r—r')]

es

£(q, a) = 1 — nq?

,

n(k + q) — n(k)

E(k + q) — E(k) + h(w + in) ©

We have allowed for the fact that

CVV v'> = dae Ck|VIK'D

Vf dkdk’ ¥ 4,4. = 2) dkdk’

(5.1.24)

402

5. Many-Particle Effects

(the factor 2 occurs because of the summation over spin). Expression (5.1.24) is sometimes

referred to as the Lindhard

T = 0K (3.5.20),

—— nie) = | ft,

formula.

Using the value of n(k) for

[kl 1, the fluctuations of the potential they produce are small, the self-

consistent-potential (random-phase) approximation is justified. An inspection of

(5.1.45)

shows

that

this

corresponds

the correlations of individual description is ineffective.

to

high

densities.

When

particles play a decisive role and

Q due to the perturbation is a

sser an

CN) = ZN gna Om =

N.

Ep

2

— Wm)(Ng +nlin) Ee — h(w

(5.1.72)

’

where we have taken account of the fact that (N _ g)am = (Ng)ma- The dielectric function (permittivity) is the ratio of the external charge density v, to the total charge density v, + . Since div D = 4nv, div E = 4n(v + (N)), this definition coincides with the usual expression D, = &(q, w)E,(D, = — 4nigev,/q? ‘

E, = — 4nige(v, + (N,))/9?). Equation (5. i 72) yields

1 €(q, @)

=14+8e

N,

4ne?

214%] Yq WwW,

2D, E,—E,

n

—

Wa

(5.1.73)

2

orm | Ng)mal” -

Expression (5.1.73) is formally exact, but has little use in particular computations, inasmuch as it requires a knowledge of the exact eigenfunctions and the energies of the many-electron system. On the other hand, the formula allows us to derive some important general relations. We calculate Im {1/e(g, w)} ane

m{1/e(q,2)}= 3D (Hy— Wm) ICN nn? x 6(E, — Eq — hw) =

2 4n7e?

En

Fe be7(-E5)

x [ exp( — FF) Jl m2 aUE, — Eq — hw) _

4n7e?

7

lex

PL

ho

er)

% [(Ng mal? 5(Eq — Em — hoo) .

(LeP\

E,,

— ar

(5.1.74)

5.1.

Plasma Phenomena. Screening

413

The expression on the right-hand side of (5.1.74) can be related to the density correlator that was introduced in Sect. 2.7.1:

S(q,w) = f deel Ni(t)N_4(0)> =f

©

-@

dre" wal Nene

(Em ~ ea |

(5.1.75)

X(N q)am = 208Y. Wul(Ne wn? 5(Eq — Eq — eo) , hq?

1

5(4,0) = — 57021 — exp(—ha/keT)| m{1/e(g,o)} . Equation (5.1.76) expresses the content of what is known dissipation theorem [5.12].

(5.1.76) as the fluctuation-

Now we can express the scattering probability of fast electrons in terms of

permittivity. Substituting v, = 4xe?/g? and (5.1.76) into (2.7.15) yields 4

~

2\2

mefig?

1 — exp(— hiw/kyT)

m{1/e(q,w)} ,

(5.1.77)

w = [E(k)— E(k +q)T/h . In the absence of damping, we write ¢(g, w) > €(g, w) + id, 6 > +0, Im {1/e(g, a)} = — 2d(e(g, @)) .

(5.1.78)

Substituting (5.1.78) into (5.1.77), we find

R =

8x20? __

5 (ng + 1)6(e(q,)) ,

hq

(5.1.79)

where

_ 1 i ” ~ exp(hw/kpT) — 1

(5.1.80)

is the Planck distribution function. The occurrence of the 6 function (5.1.79)

indicates that the scattering is accompanied

by creation of a plasmon and

conforms to (5.1.58); the occurrence of the representative term n,, + 1 (the same

414

5. Many-Particle Effects

as that for phonons) signifies that plasmons obey Bose-Einstein statistics. Incidentally, since the quantity hw, is normally approximately equal to 10 eV, we may always assume that n,, ~ 0. A very important relationship exists between the permittivity and the freeenergy of a many-electron system. We represent the electron-electron inter-

action Hamiltonian as Hin

_ 2ne?

= IT

7

=

q

~ «#0

(5.1.81)

V_y-N),

where we use the expression NJN_,=N~'Y,,expLig:(r; — rI—2N. The a free energy is equal to

F = —kgTinz= = hy Tin Trexp( = orkTee) , with #%, being the Hamiltonian

of noninteracting electrons. Making

placement e? > Ae? in (5.1.81), we calculate 6F(A)/dd:

OF __ikpT OZ(A)_

da ~~

1

2(a) aA = Za

= (Hua

.

| Mme

(5.1.82) the re-

_ Hot A Hin

kT

>

(5.1.83)

where ¢ ... >, is the averaging over the Gibbs ensemble with the replacement e? — Je*. We then substitute (5.1.81) into (5.1.83), allowing for the fact that x CNN g> = f dS°S(qa)

(5.1.84)

Here we have to invert the Fourier transform (5.1.75) and set t = 0. In turn, S(qg, w) is expressed in terms of Im {1/e(g, w)} according to (5.1.76). As a result, we find OF

a

2 dw

mY ( M3 Et — exp(— hak 2

x Im {1/e,(¢, @)} + oa) hq?

:

-

T)]*

(5.1.85)

5.2

The Fermi-Liquid Theory

415

where the replacement e? — Ae? should be made in the calculation of ¢,(g, w). For A = 0 the quantity F is known. Calculating the permittivity within some

approximation (for example, the random-phase approximation) and integrating (5.1.85) over 2 between 0 and 1, we find the free energy of a many-electron system and thus all its thermodynamic properties.

5.2 The Fermi-Liquid Theory 5.2.1 Major Postulates of the Landau Theory As is clear from Sect. 5.1, the random-phase approximation provides a suffi-

ciently complete description of the electronic system of a metal in the high-

density limit

kedp 21,

(5.2.1)

with ag being the Bohr radius. It is possible to calculate the permittivity, to find

the free energy heat capacity, spin and of an frequency and

of the system according to (5.1.85), and thereby to determine the compressibility, and other thermodynamic properties. Effects of externally applied field can be taken into account. Also, high“short-wave” properties can be described. However, the quantity

ky in metals is on the order of ag ' and so, even qualitatively, the random-phase approximation is dubious (although qualitatively embracing nearly all of the major features of many-particle coupling). Landau proposed a phenomenological

system of interacting electrons—the

also when

the binding energy

approach

Fermi-liquid

to the description

of a

theory [5.13]—applicable

is on the order of the kinetic energy

of the

particles. When applied to electrons in metals, this means that kpag ~ 1. On the other hand, Fermi-liquid theory deals only with sufficiently low-frequency and long-wave excitations

ho

(5.3.51)

obtained

for

slow

electrons

with

2p2 5

m* =m(1—«/6)~' =m(1+0/6)

,

(5.3.52)

.

(5.3.53)

where -

e?

(2ma\'!27/1

(=)

tne

1

is a dimensionless constant used in the theory of polarons, and we have set « d. The kinetic properties of large-radius intermediate and strong coupling polarons seem to be the most complicated. We now consider briefly the theory of small-radius polarons where |) ,

lq

q

(5.3.69)

= f dr|x(r—R,) 24 =e

™ i. dr|y(r)[2e""

(5.3.70)

Substituting (5.3.70) into (5.3.69), we write V,, in the form

y, Vin=i Met **(bg—b_,) , q

M

|

_4nFe?

Iql

fadrly(r)Pet" .

(5.3.71)

a

We solve the problem by the canonical transformation method; i.e., we find a unitary operator U such that the transformed Hamiltonian

#'=U#U* becomes

as simple

,

(5.3.72) in form

as possible.

In particular,

by

use of canonical

5.3

Electron-Phonon Interaction

445

transformation we try to eliminate the electron-phonon interaction term V. The

U will be sought in the form

U=exp [ix A(q)(b¢+ =)

,

(5.3.73)

where A(qg) is a matrix that depends on electronic variables, with Agm(Q)= O(n ¥ m); A,,(g) has to be found from the maximum simplicity condition for #’. We proceed to the calculation

V'=UVU* .

(5.3.74)

Obviously, the matrix U will be unitary if A,,(q) is real valued. Then, since the matrix A is diagonal, we have Um=0,

nem

Una -ero| i

Anal @)(b¢+0-0 | .

q

(5.3.75)

Therefore V,,,=0 at n¥#m, and

V= id M, explig: Resp

» Ann( P)(bp

+b |

“bj—b-gexo| iT Anwnbs+b-p) ’

(5.3.76)

.

Vary the operator By= exp] i

?

Anos

involved in (5.3.76), with performed in Sect. 2.7):

on |b respect

exp] —iY An(p)(bp+ on | »

P

to

A,,(p)

(analogous

(5.3.77)

computations

were

+

oBy

5A,,(P)

=ie

i

?

Ann( P)(bp+ bn ley

b_,)b¢

-exo| -i5 Anal p)(bp + »-»|- exo P

‘(by+ bn loses

YAn(P) ?

b-pesp| -iD A,(p)(bf+0.» | ’

=iU,[bf+b_,, bf]_UZ=i5 4).

(5.3.78) .

446

5. Many-Particle Effects

using

the

commutation

relations

for

the

Since BJ=bg for A,,(p)=0, (5.3.78) yields

operators

by and

Bg=bg+iAw(—9) -

b, (Sect. 2.8).

(5.3.79)

Similarly,

exp| iE Anal p)(bp+ bn |b-veso} -i> An Pib+-» | =b__—iA,()

P

P

-

(5.3.80)

Substituting (5.3.79, 80) into (5.3.76) yields

Vin = Von — 25, MgAna(—) explig: Ry) a

(5.3.81)

The calculations of the quantity

(Han = Une

US, «

(5.3.82)

are quite analogous, and are left as an exercise for the reader. As a result, we find

(Heian = Heyy + itidg Y AyaQ) (bg — By’) Oo

(5.3.83)

+ YAnn(—9)M,exp(ig: R,) q

Then, if we choose

Ann(—

M, —g=—L 4) har

the term

:

ig: exp(—iq'R,) .

in the Hamiltonian

( 5.3.84 )

which

is linear with respect to the phonon

y M2.

(5.3.85)

operators cancels out, since, from (5.3.71, 81, 83), we have

(65)an + Van = Hp

»

hw “g

5.3.

Electron-Phonon Interaction

447

The result is

B exp e')

ale

=

y at el Bb e+ om) . q

ep Hoy

0

_

DictDooo

ie R, be

n, m are nearest

at b.)]: neighbours, n=m,

-4,

otherwise

0

5.3.86) where

(5.3.87)

ho o¢= Me:

Thus, the canonical transformation leads us to the result that the effect of the electron-phonon interaction boils down, first, to a decrease in electron energy at each site by an amount 4 (known as the polaron shift) and, second, to the occurrence of a complicated dependence of the transfer integral 8 on phonon

operators. To understand the consequences of this, we average #%,,, over the

phonon subsystem for R, — R,, = 6 (the nearest-neighbor vector). This quantity is quite similar to the one used in Sect. 2.7.3, where we calculated the Debye— Waller factor (2.7.53, 55). The two quantities differ from each other only in the prefactors of the phonon operators. Without repeating all those calculations, we immediately put down the result (2.7.55, 56):

B = (96,,) = Bexp(—Sr) ,

(5.3.88)

M2

S;= 2Y has? [1 —cos 5] [N(w) + 4]

(5.3.89)

q

hoo Mil ~ Gag tanh P27) % Matt e088)6 -_!

Taking account of (5.3.89), we find the order of magnitude of S; to be

4

S; w 4

~ hao

cotanh

hao

2kyT

&

ho

kgT< hwo A

2kp T hay

\2

(5.3.90) ’

kp T>

ha

.

448

5. Many-Particle Effects

Thus, with S; > 1, the transfer integral decreases exponentially and, consequently, the conduction band narrows (5.3.88). In addition, the band decreases in width as the temperature is increased; i.e., the effective mass increases. In

replacing %,,, by its mean value we have disregarded the fluctuation scattering processes in the phonon subsystem. As may be shown [5.30], this scattering is rather slight at very low temperatures. However, with increase in 7, on the one hand, these scattering processes themselves build up and, on the other, the spin-

polaron effective mass m* oc |B|~'. Asa result, the narrow-band motion at some

temperature gives place to thermally activated jumps having the character of a

random diffusion process.

5.3.5 The Cooper Phenomenon

We saw in Sect. 5.3.4 that the electron-phonon interaction leads to an effective

attraction between electrons. This attraction brings about a drastic readjustment of the ground state of the electronic subsystem: a transition to a superconducting state. Our objective here is to consider a problem whose solution is indicative of this instability [5.36].

We explore the interaction of two electrons. The electron-electron interaction potential has the form V(r, —r,); both electrons are in Fermi-surface states. The Schrédinger equation for an electron pair has the form h2

- am (41 + 42,)W(ri. 72) + V(r, — 2) (r1, 62) 2,2

-(e+ where

h SE m

the energy

versa)

(5.3.91)

’

E is counted

from

the doubled

W(r,,r2), V(r, — rz) in a double Fourier series W(risr2) = ¥ expliky-r, thera) Wa, kk,

-

Fermi

energy.

Expand

(5.3.92)

Transform the second sum in (5.3.91):

os, Vaan,

=

akik;

exPlilky

Ve, — qk

+ q)r;

+ i(k, ~ g)"r2]

XPT

ry + har)

(5.3.93)

5.3.

Electron—-Phonon Interaction

449

Equating the coefficients of equal exponents (possible because they are ortho-

normal), we obtain

h2

1

Jy LAT — ke) + (42 — ke) aya, + wilale, —qki+g = EWak, .

(5.3.94)

Introduce new variables—the center-of-mass quasimomentum

x and the

quasimomentum with respect to the motion K—using the formulas

K=k,+h,,

K=4k,—-k,),

ky=8 3 4K,

k= K2

(5.3.95)

—-K.

Equation (5.3.94) involves only y with one value of x. We choose the value

« = 0. This corresponds to a pair at rest; a moving pair can be considered simply by passing to a moving frame of reference. Denoting h2

& = 3m (A? —kh),

Gp = Wek,

(5.3.96)

We obtain from (5.3.94)

1

26g t N » ViOx-q=

Eox

»

(5.3.97)

analogous to the equation considered in Sect. 4.9.1. To solve this equation, we need to analyze the form of Vy. Before we do so,

one more circumstance must be noted. Up to this point nothing has been said about the orientation of the spins of the electron pair selected. The pair should

have minimum energy, and for the two-electron system this is achieved with the

spins being antiparallel, due to the antisymmetry of the electron wave function and due to some general theorems of quantum mechanics. It follows from the fact that antiparallel spins correspond to an antisymmetric spin-wave function

and a symmetric coordinate wave function. The latter does not vanish anywhere

and can therefore describe the ground state [Ref. 1.12, Sects. 62, 63]. Thus, we consider a pair of electrons with antiparallel spins and with total momentum equal to zero. Now we consider the potential. The electron-electron attraction is effective only in a layer of thickness hwp near the Fermi surface, wp being the Debye

frequency.

Using perturbation theory, we see that the magnitudes of the corresponding energy contributions of virtual excited states fall off rapidly with increasing

450

5. Many-Particle Effects

excitation energy. The energy hwp is the minimum characteristic binding energy and, therefore, it determines the variation scale of V,.

Further, we assume that scattering occurs only in states that lie above the

Fermi surface, since scattering below the Fermi surface is prohibited by the

Pauli exclusion principle. For this to be taken into account, we must introduce the dependence of V, in (5.3.97) to include not only g but also K. By doing this it certainly ceases to be a Fourier transform of the pairwise interaction potential, but its physical significance is retained completely. A more rigorous treatment confirms this simple, although not altogether correct, mathematical approach.

Thus, we choose the phonon contribution to V, as

ye

oO

for 0 < eg, &_, < hwy

0

otherwise

(5.3.98)

,

where V > 0. Furthermore, we have to take into account, in one way or another, the screened Coulomb repulsion between electrons, which acts more or less

uniformly over the entire energy band. We model this repulsion by the following contribution to the potential: Ve=

'

U_

{0

for0

and we have, up to linear terms in v, . 2ev . J & Jo[sin(dy + wyt) + Feo NOt CO8(5o + wyt)]

=Jo {sin

+ wyt)+ j_lsin(6, +(w+oy)t)

+sin(d9 + o-oo}

(5.4.20)

.

When w = wy, the mean value of the last term is other than zero and a resonance increase in direct current through the junction occurs. The Josephson effect, in

460

5. Many-Particle Effects

particular,

allows

very

accurate

voltage

and

magnetic

field

measurements

[5.45]. We conclude this section by returning to the problem of the difference in the

properties of first-order and second-order superconductors (type I and type II).

For superconductors of the second kind the surface tension at the interface between the normal and the superconducting phase turns out to be negative [Ref. 5.46, Chap. 5]. The increase in free energy when the field penetrates into

the superconductor and thus gives rise to a normal-phase region may thus be compensated, at H > H,,, by a surface energy gain due to the formation of an interface. Similarly, starting with the normal phase at high fields, the formation of a superconducting phase is favorable at H into the excited state |1> and, accordingly, a hole has remained in the state |0> (Fig. 5.13a). The excited electron and hole can propagate independently to produce current-carrying states (Fig. 5.13b) or conjointly as

*|t

—s

—+

—

—.

||

“II

~~

—-

_

Fig. 5.13. Exciton state types: Frenkel electron-hole currentless exciton (a), electron-hole current-carrying excitation (b) (straight arrow refers to the excited state |0); undulating arrow refers to the state |1>)

5.5

Excitons

461

the excited state of the site to produce currentless states (Fig. 5.13a). In the latter case we get an energy gain equal to the magnitude of the Coulomb repulsion of electrons on the site in the state |! and in the state |0> (to a zero approximation,

the intermolecular contribution to the energy may be neglected).

This bound state of the electron and hole on one site propagating over the crystal as a whole is said to be a Frenkel exciton [5.48]. To estimate its energy

we use the Hamiltonian of a molecular crystal has the form

(5.5.1)

Hn 44D Vn H =F

In this expression # 'm is the Hamiltonian of the electrons in the mth molecule; Vn is the Hamiltonian of the intermolecular, normally dipole-dipole, interaction

p= Sede 3dy(Ry — Ry) (ds — Ro) “1k. RP IR, — Ry

(5.5.2)

where d, is the dipole moment operator of the mth molecule, and R,, is the vector of the same site (for simplicity, we consider the case of one molecule per unit cell).

To a zero approximation with respect to V, the wave function of a crystal

with an excited mth molecule may be chosen as the product of wave functions of individual molecules

Um = ot!)

yg) ,

(5.5.3)

ntm

with g®:") being the wave function of the mth molecule in the state (0, 1). The

antisymmetry effects (Pauli exclusion principle) are insignificant in the problem of interest and will therefore be neglected. This is certainly justified for excitations without spin flip (singlet excitons). By contrast, exchange effects are

important for excitations with spin flip known as triplet excitons. The matrix

elements of the Hamiltonian (5.5.1) for the functions (5.5.3) are as follows (see similar computations in Sect. 4.6.1):

LWmlHlr> = 4+ CQ pF

ogi? > = A+ vg 5

(5.5.4)

where

nl 9) 4 = (9) Halo?) — (9H is the excitation energy in the molecule. The zero of energy is the energy of the state |0), i.e., the ground state. The dipole-dipole interaction matrix element v,,

462

5. Many-Particle Effects

depends only on the difference R,, — R,. Then, just as in the electron-spectrum problem in the LCAO method (Sect. 4.5.3), we can immediately write the

spectrum and eigenfunctions E(k) = 4 + v({k) ,

(5.5.5)

Ve => Umexp(ik Ry) »

where & is the quasimomentum running over the Brillouin zone, and v(k) the Fourier transform of v,,,. Thus, the interaction between molecules expands the excited level into an exciton band. The exciton velocity is, as usual, equal to (1/A)OE(k)/dk. Excitons do not transport current but transfer energy and may show up in optical spectra [5.49-51]. Another limiting case is the bound state of an electron and hole with a radius that is much larger than the lattice constant. This case occurs frequently in semiconductors (Wannier—Mott excitons). If the electron and hole are described

in the effective-mass approximation and their interaction is described with the

help of the static dielectric permittivity, the effective Hamiltonian will be

=

h?

hi?

HH = —~——A,-~——A, 2m,

*

2m,

e?

*

where the subscripts e and h

(5.5.6)

- ———__ , Eolre— Fal

refer to electrons and holes respectively. This is

analogous to the problem of the hydrogen atom with the substitutions m — m,, M —™m, (M being the nuclear mass), e?

the expression for the spectrum

pe*

1

h? K?

+

EWR) = — 95363 2? + Dm,4+m,)

e?/e9. We can thus immediately write

n= 1,2,3,...,

(5.5.7)

where p = m,m,/(m, + m,) is the reduced mass, and K is the center-of-mass quasimomentum. The radius of the bound state at n = 1, h2

ax", pe

(5.5.8)

is large for the same reasons as in the treatment of impurity states in semiconductors (Sect. 4.5.1), i.e., the smallness of m, and, consequently, u (normally m, > m, and p= m,) and the large values of 9. The fulfillment of the inequality a> d justifies allowance for the crystalline environment by introducing static dielectric permittivity (Sect. 4.5.1). A more complicated type of elementary excitation is the biexciton, the bound

state of two Wannier—Mott excitons in analogy to the H, molecule. Biexcitons

5.6

Transition Metals and Their Compounds

463

are quite stable [5.52]. However, since holes are much lighter than protons, the energy of the zero-point oscillations of such a “molecule” is large, and therefore it is only loosely bound. In fact, the binding energy of a biexciton, expressed in

units of pe*/2h7e2, is appreciably less than that of H,, expressed in units of

me‘*/2h?. The

use of lasers permits creation of large exciton concentrations

nz 10'® cm~? in semiconductors; this is feasible when

na~l.

(5.5.9)

In this case the excitons form a liquid rather than a gas. The liquid could be similar to liquid hydrogen (i.c., it could consist of biexcitons), or to liquid alkali

metals (i.¢., it could form a two-constituent electron-hole plasma). Because of

the small binding energy of the biexciton, the latter alternative is the one

actually found [5.53]. When the condition (5.5.9) is satisfied, the bound state of

the electron and hole vanishes because of Coulomb potential screening, and an

electron-hole plasma is formed (cf. the discussion of the Mott transition in Sect. 5.1.3 and formula (5.1.44)). This formation of an electron-hole plasma results in a metal-insulator transition in a system of light-excited electric current carriers. The metallic electron—hole liquid can also form drops that can be

observed experimentally. The study of this type of liquid is a rapidly developing area of the physics of semiconductors.

5.6 Transition Metals and Their Compounds 5.6.1

Properties of d and f States

An interesting group of solids is the family of materials containing transitiongroup elements whose atoms have incomplete d or f subshells (Chap. 1). These materials can be metals, semiconductors, or insulators, and most of the solidstate concepts introduced earlier are applicable to them subject to the important restriction that many-electron effects play a much larger role in these com-

pounds. Many-electron effects are manifest most dramatically in the phenom-

enon of magnetic ordering, which actually has not ever been detected in substances not containing transition elements. Even when such materials pos-

sess no magnetic order, they still exhibit unusual thermal, magnetic, optical,

electrical, and even mechanical properties. The nature of these anomalies is due to the peculiar behavior of d and f states (to be discussed in Sect. 5.6.3).

We start by considering the atom of a transition element. There are several groups of elements that have unfilled d and f subshells. The electronic

configurations of these atoms are specified in Table 1.10 and in Sect. 3.1. We have to pay attention to the small radius of d-electron and particularly f-electron

464

5. Many-Particle Effects

subshells in comparison with characteristic solid-state distances. By character-

istic spacing we mean the distance between nearest ions in the metallic state of a relevant element. Another interesting feature is that the filling of d and f subshells proceeds in jumps at the middle and end of each series. With the atomic number changing by unity, the occupation number changes by two

electrons. The situation at the end of a series is clear—this is the usual effect of the elevated stability of a filled shell; the effect of the stability of a half-filled shell

occurs, for example, in chromium, where the configuration of the ground state is

3d°4s'. Here we are confronted with a fact that is impossible from the one-

electron standpoint since two states (3d and 4s) are partially filled. It would seem that either the energy of 4s states should be lower than that of 3d states—then the configuration of the ground state should be 3d* 4s?—or vice versa—then we should have 3d° 4s°. This indicates that the one-electron approach is inadequate to describe the atoms of transition metals. In particular, we have to take account of the exchange correlation interaction (Sect. 4.6.1), which leads to the formation of an atomic magnetic moment. As has been stated (Sect. 4.6.1), the effect comes

about because electrons with parallel spins are at larger mean distances than are electrons with antiparallel spins, because of the Pauli exclusion principle, and,

therefore, repel more weakly. If this interaction is sufficiently large compared with the characteristic one-electron excitation energies (for example, the transition from the 3d° to the 4s’ state), which are relatively small for transition elements, then it is more favorable energetically to have two partially filled states rather than have electrons with antiparallel spins. Thus, isolated atoms can possess magnetic moments. This applies not only to transition-element atoms but also to atoms with partially filled s and p states. The magnitude of the magnetic moments may be determined from Hund’s rules, according to which the total spin moment S is maximum in the ground state with a given configuration and the orbital moment L is maximum when S is

fixed [1.12]. The total atomic moment J arises from the spin-orbit coupling and

is equal to L + S for a more than half-filled subshell and is equal to |Z — S| in the opposite case. For a subshell filled exactly by half, L=0 and J =S. What happens to electronic states as atoms are united into a crystal? We are concerned primarily with the states of unfilled subshells, for these states are immediately responsible for all the properties of the crystal, apart from the ultrahigh frequency properties. As stated in Sect. 4.5.3, when atomic states form a band we have a gain in kinetic energy—the stronger the overlap of wave

functions the larger the gain—and a

loss in Coulomb repulsive energy. If the

radius of a relevant electron subshell exceeds the nearest atom or ion distance,

the gain in kinetic energy turns out to be decisive and Bloch states arise. The

interaction here should be approached in terms of the Fermi-liquid theory or

similar approaches. This situation occurs for outer s- and p-shell electrons. These electrons are either shared between all the atoms, as in metals, or between pairs of adjacent sites to form covalent bonds as in silicon-type crystals, or they fill the shells of only some of the atoms as in ionic crystals (Sect. 1.8).

5.6

Transition Metals and Their Compounds

465

A different situation occurs for the states of f electrons. These electrons

maintain their atomlike character and do not form a band (except for the so-

called intermediate valence cases [5.54]). In fact they only participate in chemical bonding indirectly, through their influence on valence electrons. The magnetic moments of f elements in compounds or in the metallic state are normally close to those of the corresponding atoms. Both cases may be realized for d states. The atom-like behavior of d states persists in many semiconducting compounds, for example NiO. In other cases metal-insulator transitions occur due to changes in the behavior of d states and due to band formation. In pure d metals, as will be seen, we deal with a “band” limit, although the latter shows some features of the atom-like localized behav-

ior of d states.

Attempts at providing a unified description of the “localized itinerant” behavior of d electrons date back to Shubin and Vonsovsky [5.55], who devel-

oped what is known as the polar model of the crystal. Despite the subsequent

effort by many physicists, the problem of constructing such a description is still far from being resolved. We start by considering localized d (or f) states. Each transition atom then possesses a magnetic moment. Electron motion, associated with a change in the

number of electrons on given lattice sites, is energetically unfavorable and the only type of low-lying energy excitations is due to the rotation of moments

relative to each other (known as spin waves). This situation is described in a model by Heisenberg [1.25], which we consider briefly in what follows.

5.6.2 The Heisenberg Model To start with, we briefly recall some of the results discussed in Sect. 1.7. We examine a system of two electrons that interact via the potential V(r —r’), which will be viewed as a perturbation. In the absence of an interaction, the electrons were in the states ¢, and ¢,. The total wave function of two noninteracting electrons should be antisymmetric with respect to the permutation of spin and spatial coordinates; on the other hand, this function is the product of the

coordinate function and spin-wave function

Wry, O15 P25 F2) = PM» F2)K(01, 92) -

(5.6.1)

Permuting o, and a, for parallel spins, with total spin S = 1, alters nothing, so X(01, 02) = x(02,0;) O(ri,72)= —Plr2,71)

, -

466

5. Many-Particle Effects

Then

1

V2

Ps = 1 (P72) = =LOi(r)G2(r2) — G1 (r2) 9201) -

(5.6.2)

For antiparallel spins, S = 0, x(o,, 02) = —x(@2, 6;), as is proved in any text-

book on quantum mechanics. Consequently, $(r,,r.) = $(r2,r,) and os =ol"1, 62) =

1

alorterdontra) + Pilr2)p2(7;))

-

(5.6.3)

When there is no interaction, the energy does not depend on the total spin. In first-order perturbation theory the energy correction due to an interaction is

AEs = fdr,dry\ps(r,, 42)? Vir, —r2) = AtJ

,

(5.6.4)

where Vir, —r2),

A=

fdr, dr. |p, (71)? o2(r2)!?

J=

fdr, dr, 93 (r,)e2(r1)e1(r2) 98 (r2) Vr, —Fr2).

(5.6.5)

The minus sign in (5.6.4) corresponds to S = 1 and the plus sign to S=0. We

introduce the total-spin operator

S=8,+48,, the square of which is equal to S(S + 1):

S(S +1) = 82 + 8 + 428, -8, = 2:44 + 1) +28, -8,

(5.6.6)

=3+28,-§. Hence the operator with the eigenvalues — 1 for S = 1 and +1 for S =0 may be represented as

P=1-—S(S+1)= —4(1+48,-8) .

(5.6.7)

Expression (5.6.4) may then be modified to

4E=A—J/2—2J5,-8,

.

(5.6.8)

Thus, the antisymmetry of the total wave function for the system of two interacting electrons leads to the energy depending on the mutual orientation of spins of the form (5.6.8) [1.26].

5.6

Transition Metals and Their Compounds

467

This coupling, called the exchange interaction, considerably exceeds the magnetic dipole-dipole interaction of spin moments, which is proportional to (v/c)? (with v being the mean velocity of electrons, and c the velocity of light in vacuum). The hypothesis as to the exchange nature of the forces responsible for magnetic ordering was propounded independently by Frenkel [5.56] and Heisenberg [1.25]. For the many-electron system the spin-dependent part of the interaction Hamiltonian is represented in the form

Hy = - Yad &

(5.6.9)

with the sum being carried out over all electrons. Expression (5.6.9) is the most general representation that can be constructed out of spin operators by taking

into account only pairwise spin interactions for spin 1/2, and §? = 3/4 and no

other functions, except for a linear function, can be constructed out of §- &.

As for expressions like (5.6.5), we should not take them too seriously. First, it is not always possible to allow for the interaction by using perturbation theory and, second, contributions to J can exist which are of a quite different nature.

We briefly consider, for example, what is known as kinetic exchange. Let two electrons be on two neighboring sites in the nondegenerate orbital state (Fig. 5.14). The repulsive interaction energy of two electrons on the same

site is larger than that of two electrons on different sites (the former exceeds the

latter by an amount U), and the matrix element of the Hamiltonian, 8, which corresponds to transitions of an electron from site to site, in the excited state, is much less than U. For the antiparallel spin orientation (Fig. 5.14a) this virtual process, taken into account in the second-order perturbation theory, then causes the energy of the system to decrease by 28?/U. The factor 2 here has appeared

because of the contribution of two electronic transitions. For the parallel orientation (Fig. 5.14b) this process is prohibited. The corresponding contribution to the exchange integral is equal to the energy difference for differing spin orientations J=E,,-£,

-

=

2p? vu

(5.6.10)

ae ! a

b

Fig. 5.14. Indirect exchange: lowering of energy as a result of virtual transitions for antiparallel spin orientation (a), in comparison with the parallel orientation (b)

468

5. Many-Particle Effects

Indeed, the exchange integral is half the energy difference of the singlet and

the triplet state J=4(Es=1

— Es=o)

.

Eg - , is the energy of the state with parallel spins E, ,. The state with antiparallel spins, may with equal probability be either a singlet or a triplet state with S, = 0.

So

Ey, = 4(Es-1

+ Es-o)

»

and therefore

J = Es-, — 4(Es-1 + Es-0) = Ey, — Ey,

-

Actually, an approximately similar exchange interaction, due to the excited

states of anions that separate transition-element ions, the indirect Kramers—

Anderson exchange also plays a leading role in transition-metal compounds. The resultant exchange integral may be both positive (5.6.5) and negative

(5.6.10).

It is normally believed that the exchange integral between d(f) electrons on different sites is independent of the orbital states in which these electrons reside. Thus, in (5.6.9) we may then carry out the sum over all electrons on each site and omit the intrasite exchange energy to obtain

J = ~uS§

(5.6.11)

‘

where the sum now is taken over the sites and § is the total spin of the d(f) shell. The model described by the Hamiltonian (5.6.11) is referred to as the Heisenberg model. Another simplification that is used frequently and which we will employ, is

the nearest-neighbors approximation: ij =

J,

if iandj are nearest neighbors

0,

in all other cases

.

(5.6.12)

If J > 0 in the model (5.6.11), then parallel ordering of all spins is the most favorable energetically. With J ? .

the

(5.6.23)

Thus, at temperatures that are lower than some temperature, called the Curie point 7;, there exists in the system a spontaneous magnetization

(5.6.24)

M = guj),

(with g being the g factor), which disappears at high temperatures. This is a second-order ferromagnet-paramagnet phase transition. We now calculate the magnetic susceptibility y at T > 7,. In an external magnetic field & the Hamiltonian will involve an extra term

#,= awh

S,

(5.6.25)

it being necessary in (5.6.18) to make the replacement x > x + y, where y= Ihe

Sh

= yn.

(5.6.26)

472

5. Many-Particle Effects

Here we have taken account of the fact that ¢$) | A = hn. Then

Hes

x=

p

_

_ 225 S? TT

B(x +

B

T s(O(x + y) = F(x ty) ’ (5.6.27)

y

*" T/Te-1’ since in sufficiently weak fields x, y

(Si, 71. = iS;ae » (St, S9]- =iS36, , (Si, jlbi; -

(5.631)

5.6

Transition Metals and Their Compounds

The operator S; increases the spin projection operator S; decreases it by unity.

473

on site i by unity, and

the

Consider the state with one spin changed:

(5.6.32)

w =Yc(i)S; 10> , i

where |0> is the ground state of the ferromagnet, and the spin projection on each site is equal to S. The equation of motion for the operator S; has the form = [S;,#]-

=

~ YInlSF

$1

.

(5.6.33)

Calculate the commutator in (5.6.33), allowing for the equations

(5.6.34)

S38z,

A

+ 5S, = 4(Sf Sy + $7 SP) and

(A, BC]_ =[A,B]_C + B[A,C]_

.

(5.6.35)

As a result, we obtain from (5.6.30, 34, 35)

(8; .§-5.1- =(87,$-5. + $187, 8,1-

= 487, S71 -Sp +087, 891-Si+48; LS; S01 + S3(S;, Si]_ = — 6,878; + 6,87 &

(5.6.36)

- buSj Si+ 64 S38, =- 6,85 (SF — bu)

+ 587 Si — 5y57 SF + 5457 (Si — 6,) Acting on the state |0), the operator $7

gives the number S at any value of i.

Here we are only interested in the case of j # k. Then

dS;

ik 7

so

sg

|0> = SY Ju ldy(S7 — S_)10> + d4(S7 — 5; )/0>]

;

= SY Iy(Si —S;)|0>.

(5.6.37

j

(Ji; = Jj). The Schrédinger equation for the function y then becomes

ih.,

OW

= BW = 28 ¥ eli). Jy(S,e- — §;¢- 10>

= 2ST JijLe(i) — e(f)1S; 10) .

(5.6.38)

474

5. Many-Particle Effects

Expression (5.6.38) is satisfied identically if

Ec(i) = 28¥. J,j[e(i) — e(i)] «

(5.6.39)

i

Since J,; depends only on R; — Rj, we, as usual, try solutions of the equation in the form of travelling waves c(i) = exp(ik-R;)

.

(5.6.40)

Here E(k) = 2S¥.J,,(1 — exp(ik-(R, — Rj))] i

-

(5.6.41)

In the nearest-neighbors approximation

E(k) = 25SY

[1 —exp(ik-d)]

,

(5.6.42)

with 6 being nearest-neighbor vectors. In the isotropic case for small k

2x [1 —exp(ik-6)] = 2 (kd? ¥

5

(k-6)

=

;

2 p2g2

3

and

E(k) = Dk?d? , ky T

D=i2JS x x Se

.

(5.6.43)

The D is called the spin-wave stiffness constant. The vanishing of E(k) with

k 0 results from the symmetry of the model (see acoustic branches of vibrations in Sect. 2.1.3). Indeed, for all spins to be rotated through the same angle, no energy needs to be expended; therefore, to rotate site spins through angles that differ as little as desired [A — 0 when c(i) ~ const], it is necessary to spend an arbitrarily small amount of energy.

In an external magnetic field or in taking account of a weak spin-orbit

interaction (i.e., when the energetically most favorable spin directions with respect to crystallographic axes arise—magnetic anisotropy), the symmetry is

5.6

Transition Metals and Their Compounds

475

perturbed and a gap emerges in the spin-wave spectrum. Normally the gap is

small compared to kg T and will therefore be disregarded.

At low temperatures (7 < 7.) the number of spin waves is small and on the

right-hand side of (5.6.30) we may set $7 x S. Then (2S) !/2$7, (28)~ /2$7

satisfy Bose permutation relations, just as for phonons and, according to (5.6.34), spin waves are created by quasi-Bose operators. At low temperatures the spinwave distribution function is therefore defined by the Planck formula (2.1.48,

2.7.36): —

1

(5.6.44)

Ne = exp(E(h)/keT)—1*

The total number of spin waves defines the deviation of the magnitude of magnetization from its saturation value at T = 0 K:

(St) Ss

«

~ T=

a

Ng dM =

~ (2n)3S

fdkN,

.

(5.6.45)

At low temperatures (D > kg T) small k are essential, so we may exploit (5.6.43)

and extend the integration over all k. Then

d? f dk N, = 4n | (kd)?d(kd)Cexp(D(kd)?/k 7) — 17? 0

= an( 27)

3/2 w

D

2

|

o expx*

—1

.

(5.6.46)

The integral involved in (5.6.46) is equal to 2

dxx?

pe yl!

J exp x? —

am

~ Wo

=22,

0

ytite

J ayy

“fen rag? F7 yan = MEI, ‘ 4 with ¢(x) being the Riemann ¢ function. Allowance for (5.6.45, 46) yields «S$?»

77

_

T

--c(7 Cc Te )

\3/2

:

(5.6.47)

where C is a numerical factor that depends on S. This formula was first derived by Bloch [5.27], and gives a fairly good fit to experimental data for many

ferromagnets. The self-consistent field approximation, as is seen from the solution to (5.6.2), would have given an exponentially small value for the quantity (5.6.47) at T+ 0 K, contradicting experiment.

476

5. Many-Particle Effects

5.6.3 d Metals We

have stated previously

that electrons in d metals

occupy, in a way, an

intermediate position between localized d electrons in magnetic insulators and

normal itinerant electrons. On the whole, the problem as to the nature of d states is solved rather in favor of the band model [5.58, 62]. First, direct X-ray, electron, and optical spectroscopy data that provide straightforward information on energy-spectrum structure indicate the existence of a d band in d metals. Second, d metals normally exhibit large values for the heat-capacity, a fact which may be explained only by taking into account the large density of d electron states. Therefore, we have to assume that d electrons form a Fermi gas or, equally, a Fermi liquid, since the heat capacity of a localized subsystem is not

proportional to the first power of temperature, but is proportional to the number of spin waves, i.e., 7?/?. Third, direct studies of the Fermi surface in d metals with the help of the de Haas-van Alphen effect and other methods indicate that d metals possess a highly complicated structure, which is very difficult to understand unless allowance is made for the delocalization of d states.

The arguments given above apply to all d metals, both to those which possess a magnetic order and to those which do not. Of the 24 pure d elements, two elements (Cr and Mn) are antiferromagnets and three elements (Fe, Co, Ni) are ferromagnets. These prominent properties will be discussed later, whereas at this point we note that the paramagnetism of the other d metals is definitely inconsistent with the localized model. At the same time, this paramagnetism is

often dissimilar to Pauli paramagnetism, indicating that a purely band description of d bands is, nevertheless, not exhaustive.

The d band and other corresponding energy bands exhibit a number of

specific features. First and foremost, because of the comparatively weak overlap of d-type wave functions, the band is rather narrow. Accordingly, the density of d states is very large. A typical feature is that the energy dependence of the density of states is a two-peaked curve, portrayed in Fig. 5.18 together with the

corresponding curve for s states. This two-hump dependence is explained with the help of the following simple considerations [5.63] (see also Sect. 1.7). We consider a diatomic molecule with a potential such as that pictured in Fig. 5.19a. If the potential barrier between two wells is sufficiently large, the g(E)

d

E

Fig. 5.18 Density-of-electron-states function g(e) in transition-d metals, for d and s bands

5.6

Transition Metals and Their Compounds

477

Uw

a

b

Fig. 5.19. Definition of bonding and antibonding states

zero-approximation function may be chosen in the form ¢,,2(x) = g(x + a),

with (x) being the wave function of an electron with energy E, in a well.

Allowance for the overlap leads to the level E, being split up into two levels (Fig. 5.19b). The lower level has a symmetric wave function with respect to spatial coordinates and a maximum between the atomic centers (the bonding orbital). To the upper level corresponds an antisymmetric function (the anti-

bonding orbital).

The separation of states into bonding and antibonding ones is rather typical of narrow bands, which are described by the LCAO approximation (Sect. 4.6.3).

In the crystal the levels E, (Fig. 5.19b) smear into two density-of-states peaks. As regards the splitting of d states in the crystalline field, which is sometimes propounded as an explanation of the two-peaks, band calculations show that this splitting is small and is much less important than the separation of states into bonding and antibonding ones. The filling of the first peak (bonding states) is accompanied by an energy gain compared to isolated atoms, and the filling of

the second peak involves a loss in energy. This accounts for the well-known fact that the bonding energy (and, accordingly, refractive index) is increased in the

middle of each d transition series. Incidentally, such a correlation between d-

band filling and bonding energy also indicates that the contribution electrons to chemical bonding is large and d electrons are itinerant.

of d

The complicated structure of the density of d states is influenced also by the

energy overlap of these states with the s states of conduction electrons. As we

saw in connection with the weak coupling approximation (Sect. 4.3), an abrupt

spectrum readjustment, or hybridization of states, occurs near the intercept of the energy curves (Fig. 5.20). This readjustment has a large effect on the density of states. As an example, Fig. 5.21 presents the calculated density of states for molybdenum [5.64]. We now discuss the problem of the temperature dependence of the magnetic susceptibility y in d transition metals. This dependence is very unusual. For a number of d metals the magnetic susceptibility decreases rather appreciably with

increasing temperature, for other d metals it increases, and for Pd it even exhibits a maximum at some temperature.

478

5. Many-Particle Effects

Fig. 5.20. Hybridization of s and d states

40JO 20 0

Fig. 5.21. Density of electron states as a function of energy g(s) in Mo according to

0

band calculation

Some of the features peculiar to the x versus T curve may be understood on the basis of the formula (3.5.87) for the Pauli susceptibility with an arbitrary

dispersion relation. Here we rewrite the Pauli susceptibility in a somewhat different form:

w(T) = zo

1 — Ekg

ae

=lo

|

(5.6.48)

Since the function g(e) is large and varies abruptly with varying ¢, the second term in (5.6.48) may be very large indeed and may be both positive and negative.

Yet this formula cannot account for, say, the non-monotonic x( 7) relation in Pd because the correlation of d electrons plays a leading part here. In the simplest version it is taken into account in terms of Fermi-liquid enhancement

(5.2.35), by introducing

the temperature

denominator of this formula:

u(T)= tT)

1! Bowel Taal)

;

dependence

in the numerator

and

(5.6.49)

5.6

Transition Metals and Their Compounds

479

When 1! + B, 0. In a paramagnet the spin waves are known

as paramagnon. They attenuate strongly and are therefore not quasiparticles in the usual sense, but they do contribute to the thermodynamics. According to

(5.1.85), the contribution to the free energy is made by the singularities 1 /e(g, w). In spin systems it is the g- and w-dependent susceptibility that plays a similar

role. If the singularity is not a pole on the real axis of w and no quasiparticle

corresponds to it, then it will contribute to the thermodynamics. It is these paramagnons that are associated with the peculiarities of the susceptibility x, which has a large imaginary part of w. Since the contribution of the one-electron degrees of freedom to the entropy is suppressed, the entropy is determined chiefly by the spin fluctuations and these are responsible, in particu-

lar, for the Curie-Weiss behavior. For a more detailed treatment of the various theoretical approaches to the temperature dependence of the susceptibility in d

metals, see [5.65, 66].

What has been said above applies equally to ferromagnetic d metals. The condition of instability with respect to ferromagnetism may be obtained by use of the Fermi-liquid theory, from the divergence of the susceptibility

1+B,